| L(s) = 1 | + 37.2·2-s + 132.·3-s + 873.·4-s + 4.92e3·6-s + 6.81e3·7-s + 1.34e4·8-s − 2.19e3·9-s − 4.07e4·11-s + 1.15e5·12-s − 2.85e4·13-s + 2.53e5·14-s + 5.36e4·16-s + 4.91e5·17-s − 8.16e4·18-s + 7.13e5·19-s + 9.01e5·21-s − 1.51e6·22-s + 1.62e6·23-s + 1.77e6·24-s − 1.06e6·26-s − 2.89e6·27-s + 5.95e6·28-s − 2.06e6·29-s + 7.21e6·31-s − 4.89e6·32-s − 5.39e6·33-s + 1.82e7·34-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.942·3-s + 1.70·4-s + 1.55·6-s + 1.07·7-s + 1.16·8-s − 0.111·9-s − 0.839·11-s + 1.60·12-s − 0.277·13-s + 1.76·14-s + 0.204·16-s + 1.42·17-s − 0.183·18-s + 1.25·19-s + 1.01·21-s − 1.38·22-s + 1.20·23-s + 1.09·24-s − 0.456·26-s − 1.04·27-s + 1.83·28-s − 0.542·29-s + 1.40·31-s − 0.824·32-s − 0.791·33-s + 2.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(10.63996462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.63996462\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 37.2T + 512T^{2} \) |
| 3 | \( 1 - 132.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 6.81e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.07e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.91e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.13e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.62e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.06e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.21e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.32e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.95e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.55e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.25e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.62e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.03e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.36e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.21e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.11e8T + 7.60e17T^{2} \) |
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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
−10.23514023316964934063074088238, −8.986992918936500149823018062656, −7.87019847704877231763975855068, −7.30193478478467943268903389549, −5.65650136278658996180009307068, −5.19385908957625374843823774845, −4.09367110296502992541573069986, −3.00315286935723119907264886605, −2.50260876457199792687854383766, −1.12309451172451028859462902341,
1.12309451172451028859462902341, 2.50260876457199792687854383766, 3.00315286935723119907264886605, 4.09367110296502992541573069986, 5.19385908957625374843823774845, 5.65650136278658996180009307068, 7.30193478478467943268903389549, 7.87019847704877231763975855068, 8.986992918936500149823018062656, 10.23514023316964934063074088238
