| L(s) = 1 | + 2-s − 3-s + 4-s + 0.561·5-s − 6-s − 5.12·7-s + 8-s + 9-s + 0.561·10-s + 11-s − 12-s + 4.56·13-s − 5.12·14-s − 0.561·15-s + 16-s − 3.12·17-s + 18-s − 4·19-s + 0.561·20-s + 5.12·21-s + 22-s − 24-s − 4.68·25-s + 4.56·26-s − 27-s − 5.12·28-s + 4.56·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.251·5-s − 0.408·6-s − 1.93·7-s + 0.353·8-s + 0.333·9-s + 0.177·10-s + 0.301·11-s − 0.288·12-s + 1.26·13-s − 1.36·14-s − 0.144·15-s + 0.250·16-s − 0.757·17-s + 0.235·18-s − 0.917·19-s + 0.125·20-s + 1.11·21-s + 0.213·22-s − 0.204·24-s − 0.936·25-s + 0.894·26-s − 0.192·27-s − 0.968·28-s + 0.847·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.928483844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928483844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 7.43T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−8.569760112048533711495903405879, −7.39399973736778786757092821524, −6.48162928746938984919413204371, −6.29370509111723728189056254365, −5.78392269857247572995609114421, −4.53681080622085118219248382046, −3.90278781533997207699035937457, −3.13267518387255519220683094336, −2.16369845345674874245726643580, −0.71804524588869491190238587920,
0.71804524588869491190238587920, 2.16369845345674874245726643580, 3.13267518387255519220683094336, 3.90278781533997207699035937457, 4.53681080622085118219248382046, 5.78392269857247572995609114421, 6.29370509111723728189056254365, 6.48162928746938984919413204371, 7.39399973736778786757092821524, 8.569760112048533711495903405879
