| L(s) = 1 | − 13.2·2-s + 46.3·4-s + 17.9·5-s + 1.07e3·8-s − 236.·10-s − 3.12e3·11-s − 1.42e4·13-s − 2.01e4·16-s − 5.50e3·17-s − 2.90e4·19-s + 832.·20-s + 4.13e4·22-s + 3.99e3·23-s − 7.78e4·25-s + 1.88e5·26-s − 1.48e5·29-s + 2.32e5·31-s + 1.28e5·32-s + 7.27e4·34-s − 2.15e5·37-s + 3.83e5·38-s + 1.93e4·40-s + 7.16e5·41-s − 1.57e5·43-s − 1.45e5·44-s − 5.27e4·46-s − 1.08e6·47-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.362·4-s + 0.0642·5-s + 0.744·8-s − 0.0749·10-s − 0.708·11-s − 1.79·13-s − 1.23·16-s − 0.271·17-s − 0.970·19-s + 0.0232·20-s + 0.827·22-s + 0.0684·23-s − 0.995·25-s + 2.09·26-s − 1.12·29-s + 1.40·31-s + 0.692·32-s + 0.317·34-s − 0.701·37-s + 1.13·38-s + 0.0477·40-s + 1.62·41-s − 0.302·43-s − 0.256·44-s − 0.0798·46-s − 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1997572542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1997572542\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 13.2T + 128T^{2} \) |
| 5 | \( 1 - 17.9T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.42e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.50e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.90e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.99e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.32e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.15e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.13e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.71e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.54e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.77e6T + 8.07e13T^{2} \) |
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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
−9.829242158137397421830348779277, −9.211970700686040739358044566818, −8.071198135820571835672333069528, −7.59720168952079278124955058318, −6.53428809065482817985124483822, −5.15630820564008219360029119873, −4.30945135430655376632543206249, −2.63135015416524498018939236603, −1.74462947878161731546963227249, −0.23337558704636953355927449951,
0.23337558704636953355927449951, 1.74462947878161731546963227249, 2.63135015416524498018939236603, 4.30945135430655376632543206249, 5.15630820564008219360029119873, 6.53428809065482817985124483822, 7.59720168952079278124955058318, 8.071198135820571835672333069528, 9.211970700686040739358044566818, 9.829242158137397421830348779277
