| L(s) = 1 | + 4·2-s + 9·3-s − 16·4-s + 36·6-s − 225·7-s − 192·8-s + 81·9-s − 434·11-s − 144·12-s + 613·13-s − 900·14-s − 256·16-s − 878·17-s + 324·18-s − 731·19-s − 2.02e3·21-s − 1.73e3·22-s + 2.85e3·23-s − 1.72e3·24-s + 2.45e3·26-s + 729·27-s + 3.60e3·28-s − 7.58e3·29-s + 2.17e3·31-s + 5.12e3·32-s − 3.90e3·33-s − 3.51e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.73·7-s − 1.06·8-s + 1/3·9-s − 1.08·11-s − 0.288·12-s + 1.00·13-s − 1.22·14-s − 1/4·16-s − 0.736·17-s + 0.235·18-s − 0.464·19-s − 1.00·21-s − 0.764·22-s + 1.12·23-s − 0.612·24-s + 0.711·26-s + 0.192·27-s + 0.867·28-s − 1.67·29-s + 0.406·31-s + 0.883·32-s − 0.624·33-s − 0.521·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 + 225 T + p^{5} T^{2} \) |
| 11 | \( 1 + 434 T + p^{5} T^{2} \) |
| 13 | \( 1 - 613 T + p^{5} T^{2} \) |
| 17 | \( 1 + 878 T + p^{5} T^{2} \) |
| 19 | \( 1 + 731 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2850 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7582 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2175 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9310 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12040 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1121 T + p^{5} T^{2} \) |
| 47 | \( 1 - 29878 T + p^{5} T^{2} \) |
| 53 | \( 1 - 5740 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5174 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38717 T + p^{5} T^{2} \) |
| 67 | \( 1 - 31707 T + p^{5} T^{2} \) |
| 71 | \( 1 - 64472 T + p^{5} T^{2} \) |
| 73 | \( 1 + 19790 T + p^{5} T^{2} \) |
| 79 | \( 1 + 105000 T + p^{5} T^{2} \) |
| 83 | \( 1 + 3318 T + p^{5} T^{2} \) |
| 89 | \( 1 + 65376 T + p^{5} T^{2} \) |
| 97 | \( 1 + 919 p T + p^{5} T^{2} \) |
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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
−13.23251187252445950738092747586, −12.51176547806594565788689298074, −10.68503064048696314194989735062, −9.448185873574627624509837717143, −8.611816924315691906616001004545, −6.84891562773903167591976037816, −5.58257757096168098438234938674, −3.90517933596371385004437621219, −2.88777009363144954235238518329, 0,
2.88777009363144954235238518329, 3.90517933596371385004437621219, 5.58257757096168098438234938674, 6.84891562773903167591976037816, 8.611816924315691906616001004545, 9.448185873574627624509837717143, 10.68503064048696314194989735062, 12.51176547806594565788689298074, 13.23251187252445950738092747586
