| L(s) = 1 | + 2-s + 3.25·3-s + 4-s − 3.15·5-s + 3.25·6-s − 2.38·7-s + 8-s + 7.61·9-s − 3.15·10-s − 1.54·11-s + 3.25·12-s − 5.96·13-s − 2.38·14-s − 10.2·15-s + 16-s + 1.43·17-s + 7.61·18-s + 0.727·19-s − 3.15·20-s − 7.77·21-s − 1.54·22-s − 23-s + 3.25·24-s + 4.97·25-s − 5.96·26-s + 15.0·27-s − 2.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.88·3-s + 0.5·4-s − 1.41·5-s + 1.33·6-s − 0.901·7-s + 0.353·8-s + 2.53·9-s − 0.998·10-s − 0.465·11-s + 0.940·12-s − 1.65·13-s − 0.637·14-s − 2.65·15-s + 0.250·16-s + 0.348·17-s + 1.79·18-s + 0.166·19-s − 0.706·20-s − 1.69·21-s − 0.329·22-s − 0.208·23-s + 0.665·24-s + 0.995·25-s − 1.17·26-s + 2.89·27-s − 0.450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 0.727T + 19T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 - 1.79T + 59T^{2} \) |
| 61 | \( 1 - 2.45T + 61T^{2} \) |
| 67 | \( 1 + 0.742T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 0.198T + 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 8.45T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.75838091212570062463084648541, −7.18634942266059968750228540317, −6.69682104629004714582061937756, −5.26575198613141437200693507691, −4.53574292431290747844219608680, −3.79314410904692293177304591817, −3.24929787143912425952806842202, −2.75959917661643882765068225146, −1.80964586587144028256735644396, 0,
1.80964586587144028256735644396, 2.75959917661643882765068225146, 3.24929787143912425952806842202, 3.79314410904692293177304591817, 4.53574292431290747844219608680, 5.26575198613141437200693507691, 6.69682104629004714582061937756, 7.18634942266059968750228540317, 7.75838091212570062463084648541
