| L(s) = 1 | − 0.414·2-s − 3-s − 1.82·4-s − 5-s + 0.414·6-s + 4.82·7-s + 1.58·8-s + 9-s + 0.414·10-s + 1.82·12-s − 5.65·13-s − 1.99·14-s + 15-s + 3·16-s + 6.82·17-s − 0.414·18-s + 1.17·19-s + 1.82·20-s − 4.82·21-s − 4·23-s − 1.58·24-s + 25-s + 2.34·26-s − 27-s − 8.82·28-s − 0.828·29-s − 0.414·30-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s + 1.82·7-s + 0.560·8-s + 0.333·9-s + 0.130·10-s + 0.527·12-s − 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.750·16-s + 1.65·17-s − 0.0976·18-s + 0.268·19-s + 0.408·20-s − 1.05·21-s − 0.834·23-s − 0.323·24-s + 0.200·25-s + 0.459·26-s − 0.192·27-s − 1.66·28-s − 0.153·29-s − 0.0756·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008641514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008641514\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−9.412082325762997804166663668694, −8.128488586996754237503935423550, −7.928497381202798979266300606782, −7.25634309139352639933884134104, −5.76355159345988795424692589823, −4.95841773275613071423954266744, −4.68350005169791020883611443321, −3.57082737147522662215961429392, −1.92598098534134565080600944538, −0.77040743907051245249807996498,
0.77040743907051245249807996498, 1.92598098534134565080600944538, 3.57082737147522662215961429392, 4.68350005169791020883611443321, 4.95841773275613071423954266744, 5.76355159345988795424692589823, 7.25634309139352639933884134104, 7.928497381202798979266300606782, 8.128488586996754237503935423550, 9.412082325762997804166663668694
