| L(s) = 1 | + 42.6·2-s + 65.1·3-s + 1.30e3·4-s + 2.39e3·5-s + 2.78e3·6-s − 2.43e3·7-s + 3.39e4·8-s − 1.54e4·9-s + 1.02e5·10-s − 7.31e4·11-s + 8.52e4·12-s − 1.53e5·13-s − 1.03e5·14-s + 1.56e5·15-s + 7.79e5·16-s + 2.30e5·17-s − 6.58e5·18-s + 9.82e3·19-s + 3.13e6·20-s − 1.58e5·21-s − 3.12e6·22-s + 1.66e6·23-s + 2.21e6·24-s + 3.77e6·25-s − 6.52e6·26-s − 2.28e6·27-s − 3.18e6·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.464·3-s + 2.55·4-s + 1.71·5-s + 0.876·6-s − 0.382·7-s + 2.93·8-s − 0.784·9-s + 3.22·10-s − 1.50·11-s + 1.18·12-s − 1.48·13-s − 0.721·14-s + 0.795·15-s + 2.97·16-s + 0.668·17-s − 1.47·18-s + 0.0173·19-s + 4.37·20-s − 0.177·21-s − 2.83·22-s + 1.24·23-s + 1.36·24-s + 1.93·25-s − 2.80·26-s − 0.828·27-s − 0.977·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.436176236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.436176236\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 1.87e6T \) |
| good | 2 | \( 1 - 42.6T + 512T^{2} \) |
| 3 | \( 1 - 65.1T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.39e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.43e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.53e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.82e3T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.66e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.59e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 2.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.31e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.42e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.11e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.40e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.37e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.19e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.39e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.12e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.14e8T + 7.60e17T^{2} \) |
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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
−14.28809476493711289861202980879, −13.20969895165213249360213713115, −12.68258595983280297603724157231, −10.91642325101686675819146741100, −9.654579331878349925500059602118, −7.35089461084296773587546905281, −5.77315867629032717919093823746, −5.16062527305001112102900828952, −2.91154808235261659340788383305, −2.28002038434254740173908650716,
2.28002038434254740173908650716, 2.91154808235261659340788383305, 5.16062527305001112102900828952, 5.77315867629032717919093823746, 7.35089461084296773587546905281, 9.654579331878349925500059602118, 10.91642325101686675819146741100, 12.68258595983280297603724157231, 13.20969895165213249360213713115, 14.28809476493711289861202980879
