| L(s) = 1 | − 0.645·2-s + 3-s − 1.58·4-s + 2.70·5-s − 0.645·6-s + 4.18·7-s + 2.31·8-s + 9-s − 1.74·10-s + 1.75·11-s − 1.58·12-s + 2.14·13-s − 2.70·14-s + 2.70·15-s + 1.67·16-s − 17-s − 0.645·18-s − 3.30·19-s − 4.27·20-s + 4.18·21-s − 1.13·22-s + 1.49·23-s + 2.31·24-s + 2.29·25-s − 1.38·26-s + 27-s − 6.63·28-s + ⋯ |
| L(s) = 1 | − 0.456·2-s + 0.577·3-s − 0.791·4-s + 1.20·5-s − 0.263·6-s + 1.58·7-s + 0.817·8-s + 0.333·9-s − 0.551·10-s + 0.528·11-s − 0.457·12-s + 0.594·13-s − 0.722·14-s + 0.697·15-s + 0.418·16-s − 0.242·17-s − 0.152·18-s − 0.757·19-s − 0.956·20-s + 0.914·21-s − 0.241·22-s + 0.311·23-s + 0.472·24-s + 0.459·25-s − 0.271·26-s + 0.192·27-s − 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.701690189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.701690189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.645T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 - 0.397T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 9.59T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.481990726631960495754396600319, −8.024802197089876796576441429247, −7.20351996913851214848060396219, −6.13261343889799812419079860628, −5.47387860978451924250499674049, −4.48559368478316325602798316330, −4.16142995241276812437896359190, −2.69209619515534891294703671929, −1.72764292520506146043232444574, −1.13912480118605474533249668460,
1.13912480118605474533249668460, 1.72764292520506146043232444574, 2.69209619515534891294703671929, 4.16142995241276812437896359190, 4.48559368478316325602798316330, 5.47387860978451924250499674049, 6.13261343889799812419079860628, 7.20351996913851214848060396219, 8.024802197089876796576441429247, 8.481990726631960495754396600319
