| L(s) = 1 | + 487.·2-s + 6.56e3·3-s + 1.06e5·4-s + 3.20e6·6-s − 5.16e6·7-s − 1.17e7·8-s + 4.30e7·9-s − 6.80e8·11-s + 7.01e8·12-s + 3.79e9·13-s − 2.52e9·14-s − 1.97e10·16-s − 1.80e10·17-s + 2.10e10·18-s + 1.24e11·19-s − 3.38e10·21-s − 3.31e11·22-s − 1.22e11·23-s − 7.71e10·24-s + 1.85e12·26-s + 2.82e11·27-s − 5.52e11·28-s − 2.21e12·29-s − 2.29e12·31-s − 8.09e12·32-s − 4.46e12·33-s − 8.81e12·34-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.816·4-s + 0.778·6-s − 0.338·7-s − 0.247·8-s + 0.333·9-s − 0.956·11-s + 0.471·12-s + 1.28·13-s − 0.456·14-s − 1.15·16-s − 0.628·17-s + 0.449·18-s + 1.68·19-s − 0.195·21-s − 1.28·22-s − 0.327·23-s − 0.143·24-s + 1.73·26-s + 0.192·27-s − 0.276·28-s − 0.823·29-s − 0.482·31-s − 1.30·32-s − 0.552·33-s − 0.846·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 487.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 5.16e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 6.80e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.79e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.80e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.24e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 1.22e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.21e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.29e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.43e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 4.84e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.09e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.39e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.22e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.51e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.47e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.69e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 4.85e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 2.08e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.18e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.21e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.66e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.89e16T + 5.95e33T^{2} \) |
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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
−10.93137418493080969516848494450, −9.551379223529700995585981456713, −8.423697980793849858240451860533, −7.09585592606617951139081682092, −5.88262373209317872486210490175, −4.93673740627533969960922985589, −3.63398040933042190545926156357, −3.05738722959966559755692111052, −1.71472370953512815603168372939, 0,
1.71472370953512815603168372939, 3.05738722959966559755692111052, 3.63398040933042190545926156357, 4.93673740627533969960922985589, 5.88262373209317872486210490175, 7.09585592606617951139081682092, 8.423697980793849858240451860533, 9.551379223529700995585981456713, 10.93137418493080969516848494450
