| L(s) = 1 | + 2-s + 2.62·3-s + 4-s − 2.39·5-s + 2.62·6-s − 7-s + 8-s + 3.88·9-s − 2.39·10-s − 5.65·11-s + 2.62·12-s + 0.607·13-s − 14-s − 6.27·15-s + 16-s + 5.77·17-s + 3.88·18-s − 0.749·19-s − 2.39·20-s − 2.62·21-s − 5.65·22-s + 4.39·23-s + 2.62·24-s + 0.725·25-s + 0.607·26-s + 2.31·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s − 1.07·5-s + 1.07·6-s − 0.377·7-s + 0.353·8-s + 1.29·9-s − 0.756·10-s − 1.70·11-s + 0.757·12-s + 0.168·13-s − 0.267·14-s − 1.62·15-s + 0.250·16-s + 1.40·17-s + 0.915·18-s − 0.171·19-s − 0.535·20-s − 0.572·21-s − 1.20·22-s + 0.916·23-s + 0.535·24-s + 0.145·25-s + 0.119·26-s + 0.446·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.253268848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.253268848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 0.607T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 + 0.749T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 0.624T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 1.94T + 53T^{2} \) |
| 59 | \( 1 + 2.51T + 59T^{2} \) |
| 61 | \( 1 - 9.62T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 + 2.09T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.035435415927557882478448865015, −7.62684821314137723392761834408, −6.88027494993590048574255884721, −5.83492279910341519547465184156, −5.02760732674052795992411998848, −4.19266695166870973407896975850, −3.55180454339885704107559661304, −2.81576756583252836734160549785, −2.49426923976407021274259264620, −0.900725714031319550312602029078,
0.900725714031319550312602029078, 2.49426923976407021274259264620, 2.81576756583252836734160549785, 3.55180454339885704107559661304, 4.19266695166870973407896975850, 5.02760732674052795992411998848, 5.83492279910341519547465184156, 6.88027494993590048574255884721, 7.62684821314137723392761834408, 8.035435415927557882478448865015
