| L(s) = 1 | + 100·5-s − 4.48e3·13-s − 5.79e3·17-s − 2.37e4·25-s − 4.19e4·29-s − 804·37-s − 2.66e4·41-s + 2.01e5·49-s + 3.43e5·53-s + 5.41e5·61-s − 4.48e5·65-s − 1.39e6·73-s − 5.79e5·85-s + 3.23e5·89-s + 1.04e6·97-s + 3.19e6·101-s + 3.92e6·109-s − 3.53e6·113-s − 9.00e5·121-s − 4.18e6·125-s + 127-s + 131-s + 137-s + 139-s − 4.19e6·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4/5·5-s − 2.04·13-s − 1.17·17-s − 1.51·25-s − 1.72·29-s − 0.0158·37-s − 0.386·41-s + 1.71·49-s + 2.30·53-s + 2.38·61-s − 1.63·65-s − 3.58·73-s − 0.943·85-s + 0.458·89-s + 1.14·97-s + 3.10·101-s + 3.02·109-s − 2.44·113-s − 0.508·121-s − 2.14·125-s − 1.37·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6530280591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6530280591\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 201442 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 900542 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2242 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2898 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29257058 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 93182 p^{2} T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 20990 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 422727038 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 402 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 13330 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12606965698 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 596024002 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 171570 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 62659063426 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 270878 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5372154206 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 254706899938 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 696606 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 37146261826 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 563099056738 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 161598 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 520306 T + p^{6} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15778223475116646745305119295, −9.613441991467138950578467484153, −9.044296785638404302064806020460, −8.895368660223187629335166384233, −8.292644349530524979754518310904, −7.51157827293945321710959704330, −7.31389526870603525277808848317, −7.06929230213992740772112973107, −6.30441526806363367169400980362, −5.76998465584497940326008366853, −5.52476396319498297128359792570, −4.98437822224730914992276475999, −4.36477889192776850969137003509, −3.98735640287540738740763419725, −3.32006720467307154106433823749, −2.41622312318438910924623397426, −2.19515768126679645338122028676, −1.91841384637359529216069095829, −0.904533737396570597530426262032, −0.17737734877189426716292147587,
0.17737734877189426716292147587, 0.904533737396570597530426262032, 1.91841384637359529216069095829, 2.19515768126679645338122028676, 2.41622312318438910924623397426, 3.32006720467307154106433823749, 3.98735640287540738740763419725, 4.36477889192776850969137003509, 4.98437822224730914992276475999, 5.52476396319498297128359792570, 5.76998465584497940326008366853, 6.30441526806363367169400980362, 7.06929230213992740772112973107, 7.31389526870603525277808848317, 7.51157827293945321710959704330, 8.292644349530524979754518310904, 8.895368660223187629335166384233, 9.044296785638404302064806020460, 9.613441991467138950578467484153, 10.15778223475116646745305119295
