| L(s) = 1 | − 3.07·3-s + 4.23·5-s − 2.35·7-s + 6.47·9-s + 4.53·11-s + 1.76·13-s − 13.0·15-s − 5.23·17-s + 6.15·19-s + 7.23·21-s + 3.80·23-s + 12.9·25-s − 10.6·27-s − 29-s − 0.726·31-s − 13.9·33-s − 9.95·35-s − 2.47·37-s − 5.42·39-s − 7.23·41-s − 5.98·43-s + 27.4·45-s + 5.42·47-s − 1.47·49-s + 16.1·51-s + 3.76·53-s + 19.1·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s + 1.89·5-s − 0.888·7-s + 2.15·9-s + 1.36·11-s + 0.489·13-s − 3.36·15-s − 1.26·17-s + 1.41·19-s + 1.57·21-s + 0.793·23-s + 2.58·25-s − 2.05·27-s − 0.185·29-s − 0.130·31-s − 2.42·33-s − 1.68·35-s − 0.406·37-s − 0.869·39-s − 1.13·41-s − 0.912·43-s + 4.08·45-s + 0.791·47-s − 0.210·49-s + 2.25·51-s + 0.517·53-s + 2.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409390123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409390123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 31 | \( 1 + 0.726T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−9.350558255573478138067611287038, −8.960813953831706563919730183580, −6.94076326073091862897047134023, −6.70273419362828098186611615334, −6.05294168348002700696242813823, −5.44181677892899621157309013295, −4.68229720679233509453727817266, −3.38589405243573558492115869944, −1.85372429817513573843904959040, −0.940010814931652335521669495730,
0.940010814931652335521669495730, 1.85372429817513573843904959040, 3.38589405243573558492115869944, 4.68229720679233509453727817266, 5.44181677892899621157309013295, 6.05294168348002700696242813823, 6.70273419362828098186611615334, 6.94076326073091862897047134023, 8.960813953831706563919730183580, 9.350558255573478138067611287038
