| L(s) = 1 | − 1.44·2-s + 3-s + 0.0739·4-s + 3.20·5-s − 1.44·6-s + 2.77·8-s + 9-s − 4.62·10-s − 3.71·11-s + 0.0739·12-s + 1.13·13-s + 3.20·15-s − 4.14·16-s − 5.67·17-s − 1.44·18-s + 3.14·19-s + 0.237·20-s + 5.35·22-s − 4.86·23-s + 2.77·24-s + 5.30·25-s − 1.63·26-s + 27-s + 2.25·29-s − 4.62·30-s − 1.73·31-s + 0.418·32-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.577·3-s + 0.0369·4-s + 1.43·5-s − 0.587·6-s + 0.980·8-s + 0.333·9-s − 1.46·10-s − 1.12·11-s + 0.0213·12-s + 0.314·13-s + 0.828·15-s − 1.03·16-s − 1.37·17-s − 0.339·18-s + 0.722·19-s + 0.0530·20-s + 1.14·22-s − 1.01·23-s + 0.566·24-s + 1.06·25-s − 0.319·26-s + 0.192·27-s + 0.417·29-s − 0.843·30-s − 0.312·31-s + 0.0738·32-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 | \( 1 + 1.44T + 2T^{2} \) |
| 5 | \( 1 - 3.20T + 5T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 - 1.13T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 3.39T + 53T^{2} \) |
| 59 | \( 1 + 3.87T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 + 4.36T + 67T^{2} \) |
| 71 | \( 1 - 7.54T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 - 0.0474T + 79T^{2} \) |
| 83 | \( 1 + 8.03T + 83T^{2} \) |
| 89 | \( 1 + 9.30T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.009287478652370093524343346196, −7.14943915066535728956695781352, −6.50434282229821506530047371657, −5.55786930779715392615379245563, −4.95431092873262552259555531106, −4.04810433834267663750255868637, −2.85377991428892980646437023209, −2.08788756316044523366551353049, −1.45308374355944886051194253633, 0,
1.45308374355944886051194253633, 2.08788756316044523366551353049, 2.85377991428892980646437023209, 4.04810433834267663750255868637, 4.95431092873262552259555531106, 5.55786930779715392615379245563, 6.50434282229821506530047371657, 7.14943915066535728956695781352, 8.009287478652370093524343346196
