| L(s) = 1 | − 18·3-s + 36·5-s − 120·7-s + 243·9-s − 200·11-s + 284·13-s − 648·15-s + 2.67e3·17-s + 72·19-s + 2.16e3·21-s + 3.84e3·23-s + 2.65e3·25-s − 2.91e3·27-s + 1.02e4·29-s + 1.04e4·31-s + 3.60e3·33-s − 4.32e3·35-s + 1.31e4·37-s − 5.11e3·39-s + 4.16e3·41-s − 5.83e3·43-s + 8.74e3·45-s + 1.52e3·47-s − 1.48e4·49-s − 4.81e4·51-s + 9.01e3·53-s − 7.20e3·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.643·5-s − 0.925·7-s + 9-s − 0.498·11-s + 0.466·13-s − 0.743·15-s + 2.24·17-s + 0.0457·19-s + 1.06·21-s + 1.51·23-s + 0.850·25-s − 0.769·27-s + 2.25·29-s + 1.96·31-s + 0.575·33-s − 0.596·35-s + 1.57·37-s − 0.538·39-s + 0.386·41-s − 0.481·43-s + 0.643·45-s + 0.100·47-s − 0.885·49-s − 2.59·51-s + 0.440·53-s − 0.320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.161951970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.161951970\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 36 T - 1362 T^{2} - 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 120 T + 29278 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 200 T + 46406 T^{2} + 200 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 284 T + 477054 T^{2} - 284 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2676 T + 4344262 T^{2} - 2676 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4159894 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3840 T + 9416686 T^{2} - 3840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10212 T + 64228638 T^{2} - 10212 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10488 T + 84559438 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13148 T + 125908974 T^{2} - 13148 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4164 T + 216207126 T^{2} - 4164 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5832 T + 168401542 T^{2} + 5832 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1520 T + 148144670 T^{2} - 1520 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9012 T + 816689646 T^{2} - 9012 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 55096 T + 1818478886 T^{2} + 55096 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 63444 T + 2677193342 T^{2} + 63444 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 36792 T + 1148752246 T^{2} - 36792 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 37664 T + 2440628942 T^{2} - 37664 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 37836 T + 2085902966 T^{2} + 37836 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 144888 T + 10711615534 T^{2} - 144888 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 109272 T + 10472055958 T^{2} + 109272 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 32556 T + 8836559958 T^{2} + 32556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 69092 T + 18339537030 T^{2} - 69092 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.00359389713608509880562053446, −12.79371666124281603100481008981, −12.17268988636324471011314789399, −11.85670578886744585028351907432, −11.02167456656829442878365108043, −10.56952134263192328149958739235, −10.01303258504899452306644336361, −9.775500259086358110345226739184, −9.057712789370142658991617837483, −8.162960108149546524034567002016, −7.63959226899876252803901278412, −6.74679986796480425650172322280, −6.23046515841373336090381896304, −5.94551718863399724579803357205, −4.96493999963344991211105934350, −4.67674048080619283677820955390, −3.22774984087764701324512122914, −2.84686832140967506613271271512, −1.15864406779810985125676126691, −0.78535581233743446834437386588,
0.78535581233743446834437386588, 1.15864406779810985125676126691, 2.84686832140967506613271271512, 3.22774984087764701324512122914, 4.67674048080619283677820955390, 4.96493999963344991211105934350, 5.94551718863399724579803357205, 6.23046515841373336090381896304, 6.74679986796480425650172322280, 7.63959226899876252803901278412, 8.162960108149546524034567002016, 9.057712789370142658991617837483, 9.775500259086358110345226739184, 10.01303258504899452306644336361, 10.56952134263192328149958739235, 11.02167456656829442878365108043, 11.85670578886744585028351907432, 12.17268988636324471011314789399, 12.79371666124281603100481008981, 13.00359389713608509880562053446
