| L(s) = 1 | − 376·9-s + 3.97e3·25-s + 9.74e3·29-s − 1.93e4·37-s + 4.90e3·49-s − 1.02e5·53-s + 6.07e4·81-s − 2.17e5·109-s − 8.59e5·113-s + 4.76e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.05e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.54·9-s + 1.27·25-s + 2.15·29-s − 2.32·37-s + 0.291·49-s − 4.99·53-s + 1.02·81-s − 1.75·109-s − 6.32·113-s + 2.96·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 0.552·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9627250314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9627250314\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 100 p^{2} T^{2} + 1620810 p^{3} T^{4} - 100 p^{12} T^{6} + p^{20} T^{8} \) |
| good | 3 | \( ( 1 + 188 T^{2} + 2518 p^{2} T^{4} + 188 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 5 | \( ( 1 - 1988 T^{2} - 2941914 T^{4} - 1988 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 - 238484 T^{2} + 66033443094 T^{4} - 238484 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 13 | \( ( 1 + 102620 T^{2} + 180442358406 T^{4} + 102620 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 - 3176516 T^{2} + 6207640833990 T^{4} - 3176516 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 19 | \( ( 1 + 6965180 T^{2} + 22621470083910 T^{4} + 6965180 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 - 5197124 T^{2} - 13779530488026 T^{4} - 5197124 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 29 | \( ( 1 - 84 p T + 38752030 T^{2} - 84 p^{6} T^{3} + p^{10} T^{4} )^{4} \) |
| 31 | \( ( 1 + 38512508 T^{2} + 1593361592407686 T^{4} + 38512508 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4844 T + 110769870 T^{2} + 4844 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 41 | \( ( 1 - 291486692 T^{2} + 42668818382121318 T^{4} - 291486692 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 43 | \( ( 1 - 289827220 T^{2} + 61345166906496150 T^{4} - 289827220 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 47 | \( ( 1 - 125214532 T^{2} + 108950831984560326 T^{4} - 125214532 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 53 | \( ( 1 + 25548 T + 624186862 T^{2} + 25548 p^{5} T^{3} + p^{10} T^{4} )^{4} \) |
| 59 | \( ( 1 + 2171560796 T^{2} + 2126127541217907174 T^{4} + 2171560796 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2554991908 T^{2} + 2946798340587496518 T^{4} - 2554991908 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 + 488024396 T^{2} + 1139727820718602230 T^{4} + 488024396 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 71 | \( ( 1 - 3623243684 T^{2} + 9453839783823072294 T^{4} - 3623243684 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 73 | \( ( 1 - 5360407780 T^{2} + 14906829502160167398 T^{4} - 5360407780 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8273062660 T^{2} + 32346274228228887750 T^{4} - 8273062660 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 83 | \( ( 1 + 4252222460 T^{2} + 9178881665461147590 T^{4} + 4252222460 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 - 17969292068 T^{2} + \)\(14\!\cdots\!58\)\( T^{4} - 17969292068 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
| 97 | \( ( 1 - 31936325764 T^{2} + \)\(40\!\cdots\!22\)\( T^{4} - 31936325764 p^{10} T^{6} + p^{20} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.20074259920828890543746447372, −3.78466477552657613679793018833, −3.60655386099390756534230240705, −3.48771837867424901118292182491, −3.39667994024298476184317741364, −3.18074243141301193966540460726, −3.05022900448042942223374855817, −3.00235003164267259762510021897, −2.79823770843418280081205704103, −2.79432671286498880871051365287, −2.73152939542245985961205596828, −2.39758576628249233171351364436, −2.15690730704652848614825777698, −1.84631593571833622400746459977, −1.83120356721106835172599925312, −1.73008123596208384941134885305, −1.57489996440827106698045087100, −1.52723603486694393277181685699, −1.06443237788655655796310577810, −0.899129270657683973311078444915, −0.860631971043199716807766614773, −0.57730155804344736077211524830, −0.52655257589928512560142691765, −0.19546902611902508614581004376, −0.089997002104811738691488775456,
0.089997002104811738691488775456, 0.19546902611902508614581004376, 0.52655257589928512560142691765, 0.57730155804344736077211524830, 0.860631971043199716807766614773, 0.899129270657683973311078444915, 1.06443237788655655796310577810, 1.52723603486694393277181685699, 1.57489996440827106698045087100, 1.73008123596208384941134885305, 1.83120356721106835172599925312, 1.84631593571833622400746459977, 2.15690730704652848614825777698, 2.39758576628249233171351364436, 2.73152939542245985961205596828, 2.79432671286498880871051365287, 2.79823770843418280081205704103, 3.00235003164267259762510021897, 3.05022900448042942223374855817, 3.18074243141301193966540460726, 3.39667994024298476184317741364, 3.48771837867424901118292182491, 3.60655386099390756534230240705, 3.78466477552657613679793018833, 4.20074259920828890543746447372
Plot not available for L-functions of degree greater than 10.
