| L(s) = 1 | − 3·2-s + 3·3-s + 4-s − 9·6-s − 20·7-s + 21·8-s + 9·9-s − 24·11-s + 3·12-s − 74·13-s + 60·14-s − 71·16-s − 54·17-s − 27·18-s − 124·19-s − 60·21-s + 72·22-s + 120·23-s + 63·24-s + 222·26-s + 27·27-s − 20·28-s − 78·29-s + 200·31-s + 45·32-s − 72·33-s + 162·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 1/8·4-s − 0.612·6-s − 1.07·7-s + 0.928·8-s + 1/3·9-s − 0.657·11-s + 0.0721·12-s − 1.57·13-s + 1.14·14-s − 1.10·16-s − 0.770·17-s − 0.353·18-s − 1.49·19-s − 0.623·21-s + 0.697·22-s + 1.08·23-s + 0.535·24-s + 1.67·26-s + 0.192·27-s − 0.134·28-s − 0.499·29-s + 1.15·31-s + 0.248·32-s − 0.379·33-s + 0.817·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−13.32981456858560723560074827595, −12.65740363366071915442225455338, −10.78888095921408324514890390934, −9.822753550721010754951268393039, −9.064313662876839223871561775877, −7.88157395786848657337667891265, −6.75673114987651497580179203603, −4.58729234712202629633212090175, −2.53615074670168459448852382385, 0,
2.53615074670168459448852382385, 4.58729234712202629633212090175, 6.75673114987651497580179203603, 7.88157395786848657337667891265, 9.064313662876839223871561775877, 9.822753550721010754951268393039, 10.78888095921408324514890390934, 12.65740363366071915442225455338, 13.32981456858560723560074827595
