| L(s) = 1 | − 575.·2-s + 1.40e4·3-s − 1.93e5·4-s + 1.95e6·5-s − 8.10e6·6-s + 9.45e7·7-s + 4.12e8·8-s − 9.63e8·9-s − 1.12e9·10-s − 8.76e9·11-s − 2.72e9·12-s − 3.97e10·13-s − 5.43e10·14-s + 2.75e10·15-s − 1.36e11·16-s − 5.48e11·17-s + 5.54e11·18-s − 7.84e11·19-s − 3.77e11·20-s + 1.33e12·21-s + 5.04e12·22-s + 6.21e12·23-s + 5.81e12·24-s + 3.81e12·25-s + 2.28e13·26-s − 2.99e13·27-s − 1.82e13·28-s + ⋯ |
| L(s) = 1 | − 0.794·2-s + 0.413·3-s − 0.368·4-s + 0.447·5-s − 0.328·6-s + 0.885·7-s + 1.08·8-s − 0.829·9-s − 0.355·10-s − 1.12·11-s − 0.152·12-s − 1.03·13-s − 0.703·14-s + 0.184·15-s − 0.495·16-s − 1.12·17-s + 0.659·18-s − 0.557·19-s − 0.164·20-s + 0.365·21-s + 0.890·22-s + 0.719·23-s + 0.449·24-s + 0.199·25-s + 0.825·26-s − 0.755·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.95e6T \) |
| good | 2 | \( 1 + 575.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 1.40e4T + 1.16e9T^{2} \) |
| 7 | \( 1 - 9.45e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 8.76e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 3.97e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 5.48e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 7.84e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 6.21e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.24e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 8.00e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 5.13e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.91e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 5.52e14T + 1.08e31T^{2} \) |
| 47 | \( 1 + 7.21e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.08e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 7.69e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.80e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.96e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 4.83e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.94e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.26e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.30e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 5.19e17T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.33e19T + 5.60e37T^{2} \) |
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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
−18.02334814685230810254468851922, −17.06846030132405874335653820018, −14.77344314025641349310502348618, −13.32992663521585378433896349939, −10.83506697928661100763548447710, −9.129250451201655238197164013083, −7.83134970837625860706820684166, −4.97693007275000085368321307786, −2.18779837217932631972187179474, 0,
2.18779837217932631972187179474, 4.97693007275000085368321307786, 7.83134970837625860706820684166, 9.129250451201655238197164013083, 10.83506697928661100763548447710, 13.32992663521585378433896349939, 14.77344314025641349310502348618, 17.06846030132405874335653820018, 18.02334814685230810254468851922
