| L(s) = 1 | + 35.7·2-s − 81·3-s + 767.·4-s − 2.89e3·6-s − 3.89e3·7-s + 9.15e3·8-s + 6.56e3·9-s − 2.30e4·11-s − 6.22e4·12-s + 1.13e5·13-s − 1.39e5·14-s − 6.55e4·16-s − 3.69e5·17-s + 2.34e5·18-s − 1.10e6·19-s + 3.15e5·21-s − 8.24e5·22-s − 1.32e6·23-s − 7.41e5·24-s + 4.05e6·26-s − 5.31e5·27-s − 2.99e6·28-s + 6.68e4·29-s + 6.78e6·31-s − 7.03e6·32-s + 1.86e6·33-s − 1.32e7·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.49·4-s − 0.912·6-s − 0.613·7-s + 0.790·8-s + 0.333·9-s − 0.474·11-s − 0.865·12-s + 1.10·13-s − 0.970·14-s − 0.250·16-s − 1.07·17-s + 0.527·18-s − 1.93·19-s + 0.354·21-s − 0.750·22-s − 0.984·23-s − 0.456·24-s + 1.73·26-s − 0.192·27-s − 0.920·28-s + 0.0175·29-s + 1.31·31-s − 1.18·32-s + 0.274·33-s − 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 35.7T + 512T^{2} \) |
| 7 | \( 1 + 3.89e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.30e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.69e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.10e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.68e4T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.94e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.50e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.55e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.45e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.07e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.25e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.72e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.03e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.72e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.62e8T + 7.60e17T^{2} \) |
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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
−12.44124236173768679954553614656, −11.31010183394364912406951339293, −10.37094767591806787733408684199, −8.569242186869163530208330239767, −6.62727254190135865784916644503, −6.08683728957799350961508697344, −4.70473596485180715003879729157, −3.69878597594147456563449681939, −2.18683611635779712682625863168, 0,
2.18683611635779712682625863168, 3.69878597594147456563449681939, 4.70473596485180715003879729157, 6.08683728957799350961508697344, 6.62727254190135865784916644503, 8.569242186869163530208330239767, 10.37094767591806787733408684199, 11.31010183394364912406951339293, 12.44124236173768679954553614656
