| L(s) = 1 | + 2-s + 3-s + 4-s − 1.83·5-s + 6-s + 7-s + 8-s + 9-s − 1.83·10-s + 0.198·11-s + 12-s + 14-s − 1.83·15-s + 16-s + 0.445·17-s + 18-s + 1.57·19-s − 1.83·20-s + 21-s + 0.198·22-s + 9.19·23-s + 24-s − 1.63·25-s + 27-s + 28-s − 9.88·29-s − 1.83·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.820·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.579·10-s + 0.0597·11-s + 0.288·12-s + 0.267·14-s − 0.473·15-s + 0.250·16-s + 0.107·17-s + 0.235·18-s + 0.360·19-s − 0.410·20-s + 0.218·21-s + 0.0422·22-s + 1.91·23-s + 0.204·24-s − 0.327·25-s + 0.192·27-s + 0.188·28-s − 1.83·29-s − 0.334·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.885865797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.885865797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.83T + 5T^{2} \) |
| 11 | \( 1 - 0.198T + 11T^{2} \) |
| 17 | \( 1 - 0.445T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 - 9.19T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 + 8.51T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.307T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 - 3.19T + 83T^{2} \) |
| 89 | \( 1 + 5.55T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79261572200143451696934839527, −7.35222552745934501397661780352, −6.61830566302479171472209124154, −5.72191079881529229714211653126, −4.91716620149217170617591458072, −4.33137186248650870160326361102, −3.54464616770944056319722973224, −2.96924414613553573216690738465, −1.99884975386622484648327850973, −0.896666456005942111763670546796,
0.896666456005942111763670546796, 1.99884975386622484648327850973, 2.96924414613553573216690738465, 3.54464616770944056319722973224, 4.33137186248650870160326361102, 4.91716620149217170617591458072, 5.72191079881529229714211653126, 6.61830566302479171472209124154, 7.35222552745934501397661780352, 7.79261572200143451696934839527
