| L(s) = 1 | + 35.4·2-s + 81·3-s + 745.·4-s + 1.62e3·5-s + 2.87e3·6-s + 5.25e3·7-s + 8.29e3·8-s + 6.56e3·9-s + 5.75e4·10-s + 6.32e4·11-s + 6.04e4·12-s − 9.31e4·13-s + 1.86e5·14-s + 1.31e5·15-s − 8.76e4·16-s + 3.12e5·17-s + 2.32e5·18-s + 2.95e5·19-s + 1.21e6·20-s + 4.25e5·21-s + 2.24e6·22-s − 6.60e5·23-s + 6.71e5·24-s + 6.80e5·25-s − 3.30e6·26-s + 5.31e5·27-s + 3.91e6·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.45·4-s + 1.16·5-s + 0.904·6-s + 0.827·7-s + 0.716·8-s + 0.333·9-s + 1.82·10-s + 1.30·11-s + 0.841·12-s − 0.905·13-s + 1.29·14-s + 0.670·15-s − 0.334·16-s + 0.907·17-s + 0.522·18-s + 0.519·19-s + 1.69·20-s + 0.477·21-s + 2.04·22-s − 0.492·23-s + 0.413·24-s + 0.348·25-s − 1.41·26-s + 0.192·27-s + 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.997080929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.997080929\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 35.4T + 512T^{2} \) |
| 5 | \( 1 - 1.62e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.25e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.31e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.12e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.95e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.60e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.79e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.00e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.38e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.15e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.17e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 3.57e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.64e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.30e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.06e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.17e8T + 7.60e17T^{2} \) |
| 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
−11.44751086580250066260286858363, −9.976256076006101034034690057602, −9.170346475032040593123994475320, −7.69148775219944316795736473970, −6.49942434580435086990750770643, −5.54457501525427676069785670276, −4.62364450450684884619016573674, −3.50685058633982681325880179048, −2.31524378956277574853001653100, −1.43458142729025750442753680459,
1.43458142729025750442753680459, 2.31524378956277574853001653100, 3.50685058633982681325880179048, 4.62364450450684884619016573674, 5.54457501525427676069785670276, 6.49942434580435086990750770643, 7.69148775219944316795736473970, 9.170346475032040593123994475320, 9.976256076006101034034690057602, 11.44751086580250066260286858363
