| L(s) = 1 | + 2·2-s + 9.48·3-s + 4·4-s + 18.9·6-s − 8.59·7-s + 8·8-s + 62.9·9-s + 44.0·11-s + 37.9·12-s − 8.11·13-s − 17.1·14-s + 16·16-s + 87.4·17-s + 125.·18-s − 150.·19-s − 81.5·21-s + 88.1·22-s + 23·23-s + 75.8·24-s − 16.2·26-s + 341.·27-s − 34.3·28-s + 85.9·29-s + 209.·31-s + 32·32-s + 417.·33-s + 174.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.82·3-s + 0.5·4-s + 1.29·6-s − 0.464·7-s + 0.353·8-s + 2.33·9-s + 1.20·11-s + 0.912·12-s − 0.173·13-s − 0.328·14-s + 0.250·16-s + 1.24·17-s + 1.64·18-s − 1.81·19-s − 0.847·21-s + 0.854·22-s + 0.208·23-s + 0.645·24-s − 0.122·26-s + 2.43·27-s − 0.232·28-s + 0.550·29-s + 1.21·31-s + 0.176·32-s + 2.20·33-s + 0.882·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.268808705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.268808705\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 9.48T + 27T^{2} \) |
| 7 | \( 1 + 8.59T + 343T^{2} \) |
| 11 | \( 1 - 44.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.11T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 85.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 38.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 399.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 127.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 594.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 582.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 344.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 478.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 726.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 475.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 374.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 152.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 497.T + 9.12e5T^{2} \) |
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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
−9.506573866987181043293636299458, −8.414473320305772330133430166314, −8.008686594877272549958015841442, −6.81565925600683071528738091667, −6.35303971268241167837533604601, −4.75574462872200624103723735585, −3.92692273002016973451843063560, −3.25467901375067775855177097125, −2.37020768795677841908477545496, −1.30941774467702956410574907441,
1.30941774467702956410574907441, 2.37020768795677841908477545496, 3.25467901375067775855177097125, 3.92692273002016973451843063560, 4.75574462872200624103723735585, 6.35303971268241167837533604601, 6.81565925600683071528738091667, 8.008686594877272549958015841442, 8.414473320305772330133430166314, 9.506573866987181043293636299458
