| L(s) = 1 | − 0.117·2-s + 3-s − 1.98·4-s + 0.562·5-s − 0.117·6-s + 0.467·8-s + 9-s − 0.0659·10-s − 3.66·11-s − 1.98·12-s − 2.93·13-s + 0.562·15-s + 3.91·16-s + 5.92·17-s − 0.117·18-s − 2.30·19-s − 1.11·20-s + 0.429·22-s + 2.06·23-s + 0.467·24-s − 4.68·25-s + 0.344·26-s + 27-s + 8.65·29-s − 0.0659·30-s − 1.07·31-s − 1.39·32-s + ⋯ |
| L(s) = 1 | − 0.0829·2-s + 0.577·3-s − 0.993·4-s + 0.251·5-s − 0.0478·6-s + 0.165·8-s + 0.333·9-s − 0.0208·10-s − 1.10·11-s − 0.573·12-s − 0.814·13-s + 0.145·15-s + 0.979·16-s + 1.43·17-s − 0.0276·18-s − 0.528·19-s − 0.249·20-s + 0.0915·22-s + 0.429·23-s + 0.0954·24-s − 0.936·25-s + 0.0675·26-s + 0.192·27-s + 1.60·29-s − 0.0120·30-s − 0.193·31-s − 0.246·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.117T + 2T^{2} \) |
| 5 | \( 1 - 0.562T + 5T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 2.06T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 43 | \( 1 - 0.218T + 43T^{2} \) |
| 47 | \( 1 - 6.06T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 + 5.74T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.71462453732833400467247784229, −7.40021035411024210318485151803, −6.21441819767937360227975842876, −5.35292586439964365464386210802, −4.91907900448951753508093847046, −4.04028849015391391384660488168, −3.18591191353745337599486509566, −2.45564559257365568468572378303, −1.27917655819342298507962774438, 0,
1.27917655819342298507962774438, 2.45564559257365568468572378303, 3.18591191353745337599486509566, 4.04028849015391391384660488168, 4.91907900448951753508093847046, 5.35292586439964365464386210802, 6.21441819767937360227975842876, 7.40021035411024210318485151803, 7.71462453732833400467247784229
