| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s + 10.1·5-s − 8·6-s + 20.3·7-s + 8·8-s − 11·9-s + 20.3·10-s + 71.3·11-s − 16·12-s − 42·13-s + 40.7·14-s − 40.7·15-s + 16·16-s + 20.3·17-s − 22·18-s + 91.7·19-s + 40.7·20-s − 81.5·21-s + 142.·22-s − 32·24-s − 21.0·25-s − 84·26-s + 152·27-s + 81.5·28-s − 22·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 0.5·4-s + 0.912·5-s − 0.544·6-s + 1.10·7-s + 0.353·8-s − 0.407·9-s + 0.644·10-s + 1.95·11-s − 0.384·12-s − 0.896·13-s + 0.778·14-s − 0.702·15-s + 0.250·16-s + 0.290·17-s − 0.288·18-s + 1.10·19-s + 0.456·20-s − 0.847·21-s + 1.38·22-s − 0.272·24-s − 0.168·25-s − 0.633·26-s + 1.08·27-s + 0.550·28-s − 0.140·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.901715306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.901715306\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 4T + 27T^{2} \) |
| 5 | \( 1 - 10.1T + 125T^{2} \) |
| 7 | \( 1 - 20.3T + 343T^{2} \) |
| 11 | \( 1 - 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 22T + 2.43e4T^{2} \) |
| 31 | \( 1 - 296T + 2.97e4T^{2} \) |
| 37 | \( 1 + 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318T + 6.89e4T^{2} \) |
| 43 | \( 1 - 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184T + 1.03e5T^{2} \) |
| 53 | \( 1 - 91.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 500T + 2.05e5T^{2} \) |
| 61 | \( 1 - 30.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 642.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 224T + 3.57e5T^{2} \) |
| 73 | \( 1 + 210T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 815.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.24e3T + 9.12e5T^{2} \) |
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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
−9.645582813740595956351871985673, −8.785618222073313657485952162395, −7.66289569197074928551556513183, −6.65337618441228491502437275743, −6.02990543612263850384814116027, −5.17927784856347654626433937946, −4.59432108303203310397189840610, −3.30210050984986435809655227221, −1.96198321738219333673548290917, −1.06248712243162700266691420609,
1.06248712243162700266691420609, 1.96198321738219333673548290917, 3.30210050984986435809655227221, 4.59432108303203310397189840610, 5.17927784856347654626433937946, 6.02990543612263850384814116027, 6.65337618441228491502437275743, 7.66289569197074928551556513183, 8.785618222073313657485952162395, 9.645582813740595956351871985673
