| L(s) = 1 | − 2.75·2-s + 3-s + 5.57·4-s + 5-s − 2.75·6-s + 7-s − 9.82·8-s + 9-s − 2.75·10-s + 1.56·11-s + 5.57·12-s + 5.61·13-s − 2.75·14-s + 15-s + 15.8·16-s + 4.51·17-s − 2.75·18-s + 0.767·19-s + 5.57·20-s + 21-s − 4.29·22-s − 23-s − 9.82·24-s + 25-s − 15.4·26-s + 27-s + 5.57·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.78·4-s + 0.447·5-s − 1.12·6-s + 0.377·7-s − 3.47·8-s + 0.333·9-s − 0.870·10-s + 0.470·11-s + 1.60·12-s + 1.55·13-s − 0.735·14-s + 0.258·15-s + 3.97·16-s + 1.09·17-s − 0.648·18-s + 0.175·19-s + 1.24·20-s + 0.218·21-s − 0.915·22-s − 0.208·23-s − 2.00·24-s + 0.200·25-s − 3.02·26-s + 0.192·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311381482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311381482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 0.767T + 19T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−8.801657496663624835185865885616, −8.312053206925681541612769311688, −7.919093264832216236845356568306, −6.71029809743948422635973773011, −6.43123364632496094986445523953, −5.30488005645237076712252816828, −3.64834265756915431261797015507, −2.83054287412734138540921236837, −1.65029054269606322742661575835, −1.07312001994369485887054501184,
1.07312001994369485887054501184, 1.65029054269606322742661575835, 2.83054287412734138540921236837, 3.64834265756915431261797015507, 5.30488005645237076712252816828, 6.43123364632496094986445523953, 6.71029809743948422635973773011, 7.919093264832216236845356568306, 8.312053206925681541612769311688, 8.801657496663624835185865885616
