| L(s) = 1 | − 1.88·2-s + 3-s + 1.54·4-s − 5-s − 1.88·6-s − 7-s + 0.859·8-s + 9-s + 1.88·10-s − 3.71·11-s + 1.54·12-s − 4.63·13-s + 1.88·14-s − 15-s − 4.70·16-s − 3.31·17-s − 1.88·18-s − 4.98·19-s − 1.54·20-s − 21-s + 6.98·22-s − 23-s + 0.859·24-s + 25-s + 8.73·26-s + 27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.577·3-s + 0.771·4-s − 0.447·5-s − 0.768·6-s − 0.377·7-s + 0.303·8-s + 0.333·9-s + 0.595·10-s − 1.11·11-s + 0.445·12-s − 1.28·13-s + 0.503·14-s − 0.258·15-s − 1.17·16-s − 0.803·17-s − 0.443·18-s − 1.14·19-s − 0.345·20-s − 0.218·21-s + 1.48·22-s − 0.208·23-s + 0.175·24-s + 0.200·25-s + 1.71·26-s + 0.192·27-s − 0.291·28-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
| \(L(1)\) |
\(\approx\) |
\(0.5126466205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5126466205\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.88T + 2T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 - 0.814T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 + 0.798T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 - 5.24T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 4.73T + 97T^{2} \) |
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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
−9.019514544032441252048087200794, −8.067776199769541084136746477585, −7.81246425438464210741564894396, −7.05601206265075182414848806442, −6.14999278027824445531975548643, −4.77093968183955301163289419276, −4.19905530262967322768668665643, −2.70428058287666942029186968253, −2.19279775418504845499327762542, −0.51414969398253264886154700469,
0.51414969398253264886154700469, 2.19279775418504845499327762542, 2.70428058287666942029186968253, 4.19905530262967322768668665643, 4.77093968183955301163289419276, 6.14999278027824445531975548643, 7.05601206265075182414848806442, 7.81246425438464210741564894396, 8.067776199769541084136746477585, 9.019514544032441252048087200794
