| L(s) = 1 | + 14·5-s − 22·9-s − 40·11-s + 168·19-s + 71·25-s + 12·29-s + 448·31-s + 532·41-s − 308·45-s + 610·49-s − 560·55-s + 56·59-s + 364·61-s − 816·71-s − 96·79-s − 245·81-s − 3.05e3·89-s + 2.35e3·95-s + 880·99-s + 2.49e3·101-s + 1.80e3·109-s − 1.46e3·121-s − 756·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.25·5-s − 0.814·9-s − 1.09·11-s + 2.02·19-s + 0.567·25-s + 0.0768·29-s + 2.59·31-s + 2.02·41-s − 1.02·45-s + 1.77·49-s − 1.37·55-s + 0.123·59-s + 0.764·61-s − 1.36·71-s − 0.136·79-s − 0.336·81-s − 3.63·89-s + 2.54·95-s + 0.893·99-s + 2.45·101-s + 1.58·109-s − 1.09·121-s − 0.540·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.207017110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.207017110\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 14 T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 22 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )( 1 + 212 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1514050 T^{2} + p^{6} T^{4} \) |
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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
−14.23341567688346556072371859149, −13.74491274868954831624691689823, −13.15344432305783679922466421106, −12.72684468901857924646042956660, −11.76360690257731640353147582372, −11.65948162730227733586004312172, −10.76032741095208503542510690260, −10.13607602516701637794646635586, −9.865824827960262837503444583927, −9.167988092240455839130278648581, −8.561186822214237742895964016016, −7.83188575322641856916013443564, −7.27825649109209767850958297817, −6.31848747649302459891060928263, −5.65750406396750815253673918930, −5.36475860423713532666407253837, −4.37922083359893367045961308496, −2.93827029102956006361988112693, −2.53137648893827061986403806244, −0.992653905514558847956914837587,
0.992653905514558847956914837587, 2.53137648893827061986403806244, 2.93827029102956006361988112693, 4.37922083359893367045961308496, 5.36475860423713532666407253837, 5.65750406396750815253673918930, 6.31848747649302459891060928263, 7.27825649109209767850958297817, 7.83188575322641856916013443564, 8.561186822214237742895964016016, 9.167988092240455839130278648581, 9.865824827960262837503444583927, 10.13607602516701637794646635586, 10.76032741095208503542510690260, 11.65948162730227733586004312172, 11.76360690257731640353147582372, 12.72684468901857924646042956660, 13.15344432305783679922466421106, 13.74491274868954831624691689823, 14.23341567688346556072371859149
