| L(s) = 1 | + 256·4-s + 6.56e3·9-s + 1.03e4·11-s − 5.47e4·13-s + 6.55e4·16-s + 7.99e4·17-s + 5.40e5·23-s + 3.90e5·25-s + 5.89e5·31-s + 1.67e6·36-s − 2.26e6·41-s + 3.41e6·43-s + 2.64e6·44-s − 6.98e6·47-s + 5.76e6·49-s − 1.40e7·52-s − 1.30e7·53-s − 7.86e6·59-s + 1.67e7·64-s − 3.98e7·67-s + 2.04e7·68-s + 7.30e7·79-s + 4.30e7·81-s − 9.30e7·83-s + 1.38e8·92-s + 1.62e8·97-s + 6.77e7·99-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s + 0.704·11-s − 1.91·13-s + 16-s + 0.957·17-s + 1.93·23-s + 25-s + 0.638·31-s + 36-s − 0.800·41-s + 43-s + 0.704·44-s − 1.43·47-s + 49-s − 1.91·52-s − 1.65·53-s − 0.649·59-s + 64-s − 1.97·67-s + 0.957·68-s + 1.87·79-s + 81-s − 1.96·83-s + 1.93·92-s + 1.83·97-s + 0.704·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.585826568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585826568\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - p^{4} T \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 11 | \( 1 - 10319 T + p^{8} T^{2} \) |
| 13 | \( 1 + 54721 T + p^{8} T^{2} \) |
| 17 | \( 1 - 79967 T + p^{8} T^{2} \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 - 540719 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 589679 T + p^{8} T^{2} \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 + 2262241 T + p^{8} T^{2} \) |
| 47 | \( 1 + 6983806 T + p^{8} T^{2} \) |
| 53 | \( 1 + 13061761 T + p^{8} T^{2} \) |
| 59 | \( 1 + 7864606 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 + 39816433 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( 1 - 73045634 T + p^{8} T^{2} \) |
| 83 | \( 1 + 93091441 T + p^{8} T^{2} \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 - 162643199 T + p^{8} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.53637338269310450238893522765, −12.72368526241077260352333713705, −11.95416634610319609975392492572, −10.57597117770374067667485560253, −9.496157751418395355863380369904, −7.54419779467933091952842877846, −6.73552743936133766429739537324, −4.89851293834514499211300123457, −2.93068206177018866790131117877, −1.29439260531378622269924005689,
1.29439260531378622269924005689, 2.93068206177018866790131117877, 4.89851293834514499211300123457, 6.73552743936133766429739537324, 7.54419779467933091952842877846, 9.496157751418395355863380369904, 10.57597117770374067667485560253, 11.95416634610319609975392492572, 12.72368526241077260352333713705, 14.53637338269310450238893522765
