| L(s) = 1 | + 16·2-s + 256·4-s + 2.41e3·5-s − 1.53e3·7-s + 4.09e3·8-s + 3.86e4·10-s + 5.81e3·11-s − 1.35e5·13-s − 2.45e4·14-s + 6.55e4·16-s + 4.48e5·17-s + 1.30e5·19-s + 6.19e5·20-s + 9.30e4·22-s − 2.07e6·23-s + 3.89e6·25-s − 2.16e6·26-s − 3.92e5·28-s + 6.43e5·29-s − 1.94e5·31-s + 1.04e6·32-s + 7.17e6·34-s − 3.70e6·35-s + 9.46e6·37-s + 2.08e6·38-s + 9.90e6·40-s + 2.34e7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.73·5-s − 0.241·7-s + 0.353·8-s + 1.22·10-s + 0.119·11-s − 1.31·13-s − 0.170·14-s + 0.250·16-s + 1.30·17-s + 0.229·19-s + 0.865·20-s + 0.0846·22-s − 1.54·23-s + 1.99·25-s − 0.930·26-s − 0.120·28-s + 0.168·29-s − 0.0377·31-s + 0.176·32-s + 0.920·34-s − 0.417·35-s + 0.830·37-s + 0.162·38-s + 0.611·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.547007687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.547007687\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 5 | \( 1 - 2.41e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.53e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.81e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.35e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.48e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 2.07e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.43e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.94e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.46e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.34e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.80e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.14e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.02e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.42e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.83e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.51e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.30e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.10e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.31e9T + 7.60e17T^{2} \) |
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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
−9.786519433911689982784955431539, −9.562405211659225589296047751216, −7.943483154991813392966170480283, −6.89445795620799392700793392880, −5.84013763502533003157870277388, −5.43530087742591122076030713107, −4.17631339577523478572119771215, −2.75572793022163125029428570504, −2.11689852412173548414207659746, −0.924155047530396709910487651287,
0.924155047530396709910487651287, 2.11689852412173548414207659746, 2.75572793022163125029428570504, 4.17631339577523478572119771215, 5.43530087742591122076030713107, 5.84013763502533003157870277388, 6.89445795620799392700793392880, 7.943483154991813392966170480283, 9.562405211659225589296047751216, 9.786519433911689982784955431539
