| L(s) = 1 | + 2·2-s − 7.81·3-s + 4·4-s + 14.9·5-s − 15.6·6-s − 21.7·7-s + 8·8-s + 34.0·9-s + 29.8·10-s − 31.2·12-s + 44.0·13-s − 43.5·14-s − 116.·15-s + 16·16-s + 24.9·17-s + 68.0·18-s − 21.9·19-s + 59.6·20-s + 170.·21-s + 177.·23-s − 62.4·24-s + 97.5·25-s + 88.0·26-s − 54.8·27-s − 87.0·28-s + 149.·29-s − 233.·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s + 1.33·5-s − 1.06·6-s − 1.17·7-s + 0.353·8-s + 1.26·9-s + 0.943·10-s − 0.751·12-s + 0.939·13-s − 0.831·14-s − 2.00·15-s + 0.250·16-s + 0.355·17-s + 0.891·18-s − 0.265·19-s + 0.667·20-s + 1.76·21-s + 1.61·23-s − 0.531·24-s + 0.780·25-s + 0.664·26-s − 0.391·27-s − 0.587·28-s + 0.956·29-s − 1.41·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.986524116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986524116\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 7.81T + 27T^{2} \) |
| 5 | \( 1 - 14.9T + 125T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 13 | \( 1 - 44.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 538.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 78.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 772.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 537.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 179.T + 9.12e5T^{2} \) |
| 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
−11.74505140610294317038094177763, −10.74093085092891200926110504066, −10.12782331169106890458857225513, −9.047562035416996769583225171840, −7.01837806674432088795166209380, −6.10189478832302152512382953750, −5.84457870857156428599971339694, −4.60172343632328888877082899121, −2.92740406880386426868475535022, −1.05999928890979395020567088457,
1.05999928890979395020567088457, 2.92740406880386426868475535022, 4.60172343632328888877082899121, 5.84457870857156428599971339694, 6.10189478832302152512382953750, 7.01837806674432088795166209380, 9.047562035416996769583225171840, 10.12782331169106890458857225513, 10.74093085092891200926110504066, 11.74505140610294317038094177763
