| L(s) = 1 | + 2.49·2-s + 3-s + 4.21·4-s − 0.262·5-s + 2.49·6-s + 7-s + 5.52·8-s + 9-s − 0.655·10-s + 3.55·11-s + 4.21·12-s − 2.65·13-s + 2.49·14-s − 0.262·15-s + 5.34·16-s + 7.62·17-s + 2.49·18-s − 1.10·19-s − 1.10·20-s + 21-s + 8.85·22-s − 3.50·23-s + 5.52·24-s − 4.93·25-s − 6.62·26-s + 27-s + 4.21·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.577·3-s + 2.10·4-s − 0.117·5-s + 1.01·6-s + 0.377·7-s + 1.95·8-s + 0.333·9-s − 0.207·10-s + 1.07·11-s + 1.21·12-s − 0.736·13-s + 0.666·14-s − 0.0678·15-s + 1.33·16-s + 1.84·17-s + 0.587·18-s − 0.252·19-s − 0.247·20-s + 0.218·21-s + 1.88·22-s − 0.730·23-s + 1.12·24-s − 0.986·25-s − 1.29·26-s + 0.192·27-s + 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.097395092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.097395092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 5 | \( 1 + 0.262T + 5T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 7.62T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 7.01T + 31T^{2} \) |
| 37 | \( 1 + 0.847T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 4.21T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 + 2.38T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 1.22T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 6.98T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.679910306909354499250352956320, −7.82767193484341694357453735761, −7.18765059517319592343063435895, −6.36901022590300360028419971352, −5.56845221768988952680430939682, −4.85923237783015133070167989152, −3.91193929336211455808686981073, −3.51799436854902045920794885531, −2.44641556915412758887212163224, −1.52021006722839376544459731608,
1.52021006722839376544459731608, 2.44641556915412758887212163224, 3.51799436854902045920794885531, 3.91193929336211455808686981073, 4.85923237783015133070167989152, 5.56845221768988952680430939682, 6.36901022590300360028419971352, 7.18765059517319592343063435895, 7.82767193484341694357453735761, 8.679910306909354499250352956320
