| L(s) = 1 | − 14.3·2-s + 81·3-s − 306.·4-s − 1.16e3·6-s − 2.87e3·7-s + 1.17e4·8-s + 6.56e3·9-s − 2.32e4·11-s − 2.48e4·12-s + 1.12e5·13-s + 4.12e4·14-s − 1.13e4·16-s + 1.15e5·17-s − 9.40e4·18-s − 2.13e5·19-s − 2.33e5·21-s + 3.33e5·22-s − 1.83e6·23-s + 9.50e5·24-s − 1.61e6·26-s + 5.31e5·27-s + 8.82e5·28-s + 3.67e6·29-s + 8.85e6·31-s − 5.84e6·32-s − 1.88e6·33-s − 1.66e6·34-s + ⋯ |
| L(s) = 1 | − 0.633·2-s + 0.577·3-s − 0.598·4-s − 0.365·6-s − 0.453·7-s + 1.01·8-s + 0.333·9-s − 0.479·11-s − 0.345·12-s + 1.09·13-s + 0.287·14-s − 0.0432·16-s + 0.336·17-s − 0.211·18-s − 0.375·19-s − 0.261·21-s + 0.303·22-s − 1.36·23-s + 0.584·24-s − 0.692·26-s + 0.192·27-s + 0.271·28-s + 0.964·29-s + 1.72·31-s − 0.985·32-s − 0.276·33-s − 0.213·34-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 | \( 1 + 14.3T + 512T^{2} \) |
| 7 | \( 1 + 2.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.12e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.15e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.13e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.83e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.17e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.93e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.26e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.02e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.56e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.38e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.40e9T + 7.60e17T^{2} \) |
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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
−12.23085653867274349787835081157, −10.54720186748563788126217685896, −9.797603235896881168439800824533, −8.596561347885453827436033724656, −7.938128703253869421622348298933, −6.31762571012494607582891647668, −4.59904459873906248275581035880, −3.26811832345567053259374111203, −1.49047984582795746135900628536, 0,
1.49047984582795746135900628536, 3.26811832345567053259374111203, 4.59904459873906248275581035880, 6.31762571012494607582891647668, 7.938128703253869421622348298933, 8.596561347885453827436033724656, 9.797603235896881168439800824533, 10.54720186748563788126217685896, 12.23085653867274349787835081157
