| L(s) = 1 | + 65.1·2-s − 6.56e3·3-s − 1.26e5·4-s − 4.27e5·6-s − 2.28e7·7-s − 1.67e7·8-s + 4.30e7·9-s − 5.40e8·11-s + 8.32e8·12-s + 3.45e7·13-s − 1.48e9·14-s + 1.55e10·16-s + 9.43e9·17-s + 2.80e9·18-s − 2.25e9·19-s + 1.50e11·21-s − 3.51e10·22-s + 3.45e11·23-s + 1.10e11·24-s + 2.24e9·26-s − 2.82e11·27-s + 2.90e12·28-s + 5.11e11·29-s + 1.14e11·31-s + 3.21e12·32-s + 3.54e12·33-s + 6.14e11·34-s + ⋯ |
| L(s) = 1 | + 0.179·2-s − 0.577·3-s − 0.967·4-s − 0.103·6-s − 1.49·7-s − 0.353·8-s + 0.333·9-s − 0.759·11-s + 0.558·12-s + 0.0117·13-s − 0.269·14-s + 0.904·16-s + 0.328·17-s + 0.0599·18-s − 0.0304·19-s + 0.865·21-s − 0.136·22-s + 0.920·23-s + 0.204·24-s + 0.00211·26-s − 0.192·27-s + 1.45·28-s + 0.189·29-s + 0.0240·31-s + 0.516·32-s + 0.438·33-s + 0.0590·34-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 | \( 1 - 65.1T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.28e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 5.40e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.45e7T + 8.65e18T^{2} \) |
| 17 | \( 1 - 9.43e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 2.25e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 3.45e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.11e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 1.14e11T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.56e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 8.00e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 3.66e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.17e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.90e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.82e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.41e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.47e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.31e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.34e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.37e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.55e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.47e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.41e16T + 5.95e33T^{2} \) |
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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
−10.42981692471099537595006114023, −9.720284931272362596785725506928, −8.640676026909075438912797355726, −7.17448554680399153172872639723, −5.99259523938086138497092716711, −5.08972256796473616726954852460, −3.84068440469455155873782253725, −2.81089650239495840799955433321, −0.853392886968219527073079093280, 0,
0.853392886968219527073079093280, 2.81089650239495840799955433321, 3.84068440469455155873782253725, 5.08972256796473616726954852460, 5.99259523938086138497092716711, 7.17448554680399153172872639723, 8.640676026909075438912797355726, 9.720284931272362596785725506928, 10.42981692471099537595006114023
