| L(s) = 1 | + 5.39·2-s − 3·3-s + 21.1·4-s − 9.03·5-s − 16.1·6-s + 14.1·7-s + 70.7·8-s + 9·9-s − 48.7·10-s + 11·11-s − 63.3·12-s + 56.4·13-s + 76.4·14-s + 27.1·15-s + 213.·16-s + 96.2·17-s + 48.5·18-s − 129.·19-s − 190.·20-s − 42.5·21-s + 59.3·22-s − 101.·23-s − 212.·24-s − 43.2·25-s + 304.·26-s − 27·27-s + 299.·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.63·4-s − 0.808·5-s − 1.10·6-s + 0.764·7-s + 3.12·8-s + 0.333·9-s − 1.54·10-s + 0.301·11-s − 1.52·12-s + 1.20·13-s + 1.45·14-s + 0.466·15-s + 3.32·16-s + 1.37·17-s + 0.635·18-s − 1.55·19-s − 2.13·20-s − 0.441·21-s + 0.575·22-s − 0.922·23-s − 1.80·24-s − 0.346·25-s + 2.29·26-s − 0.192·27-s + 2.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.483721720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.483721720\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 5.39T + 8T^{2} \) |
| 5 | \( 1 + 9.03T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 13 | \( 1 - 56.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 470.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 495.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.7T + 2.05e5T^{2} \) |
| 67 | \( 1 - 683.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 241.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 803.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 884.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 366.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 987.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 429.T + 9.12e5T^{2} \) |
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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
−8.274979037318176593338505135489, −7.88456421808914708476145175147, −6.86102777558218313913131883786, −6.08280419672007126376579967474, −5.63323342894119467353035856744, −4.46097465287525265378713798180, −4.17820590546754044872623620374, −3.33292568677842006245597540119, −2.09271326712865803698363346443, −1.03224390523135683332923670406,
1.03224390523135683332923670406, 2.09271326712865803698363346443, 3.33292568677842006245597540119, 4.17820590546754044872623620374, 4.46097465287525265378713798180, 5.63323342894119467353035856744, 6.08280419672007126376579967474, 6.86102777558218313913131883786, 7.88456421808914708476145175147, 8.274979037318176593338505135489
