| L(s) = 1 | + 41.8·2-s + 0.232·3-s + 1.23e3·4-s − 1.79e3·5-s + 9.71·6-s + 2.40e3·7-s + 3.02e4·8-s − 1.96e4·9-s − 7.49e4·10-s + 1.74e4·11-s + 287.·12-s − 1.22e5·13-s + 1.00e5·14-s − 416.·15-s + 6.31e5·16-s + 3.31e5·17-s − 8.22e5·18-s + 7.61e5·19-s − 2.21e6·20-s + 557.·21-s + 7.27e5·22-s + 1.23e6·23-s + 7.02e3·24-s + 1.25e6·25-s − 5.12e6·26-s − 9.14e3·27-s + 2.96e6·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 0.00165·3-s + 2.41·4-s − 1.28·5-s + 0.00305·6-s + 0.377·7-s + 2.61·8-s − 0.999·9-s − 2.36·10-s + 0.358·11-s + 0.00399·12-s − 1.18·13-s + 0.698·14-s − 0.00212·15-s + 2.40·16-s + 0.963·17-s − 1.84·18-s + 1.34·19-s − 3.09·20-s + 0.000625·21-s + 0.662·22-s + 0.918·23-s + 0.00432·24-s + 0.643·25-s − 2.19·26-s − 0.00331·27-s + 0.911·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.170984437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.170984437\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 41.8T + 512T^{2} \) |
| 3 | \( 1 - 0.232T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.79e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.31e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.23e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.03e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.44e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.54e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.51e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.42e9T + 7.60e17T^{2} \) |
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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
−20.50428634652050050324328134539, −19.56118277473468770592031451318, −16.62776829643021273195390578558, −15.13086542264021621558258926276, −14.20223157513765993123281144803, −12.22064752638243949455290575120, −11.41703236658857597520016080649, −7.45667248779142603474364026934, −5.13978899189559338188682326891, −3.30068204782290139935686375941,
3.30068204782290139935686375941, 5.13978899189559338188682326891, 7.45667248779142603474364026934, 11.41703236658857597520016080649, 12.22064752638243949455290575120, 14.20223157513765993123281144803, 15.13086542264021621558258926276, 16.62776829643021273195390578558, 19.56118277473468770592031451318, 20.50428634652050050324328134539
