| L(s) = 1 | + 4·5-s + 2·7-s + 2·11-s − 13-s − 17-s + 11·25-s + 6·29-s + 2·31-s + 8·35-s + 4·37-s − 8·41-s + 4·43-s + 4·47-s − 3·49-s − 2·53-s + 8·55-s − 4·59-s + 6·61-s − 4·65-s − 8·67-s + 6·71-s + 4·73-s + 4·77-s + 12·79-s − 4·85-s − 2·89-s − 2·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.242·17-s + 11/5·25-s + 1.11·29-s + 0.359·31-s + 1.35·35-s + 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.274·53-s + 1.07·55-s − 0.520·59-s + 0.768·61-s − 0.496·65-s − 0.977·67-s + 0.712·71-s + 0.468·73-s + 0.455·77-s + 1.35·79-s − 0.433·85-s − 0.211·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.120350394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120350394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.03728561903429, −15.29583676731756, −14.70391372941421, −14.20961902959630, −13.85198886460771, −13.34181344426789, −12.70199590982396, −12.09353700060961, −11.48037547950694, −10.77677745087536, −10.28321776132704, −9.728318050942609, −9.200042326416303, −8.656213814568994, −8.000636147893708, −7.174459875719406, −6.380427008777749, −6.202644862232076, −5.205106129123606, −4.923177176383010, −4.076892369769830, −3.001065602676283, −2.304538115017910, −1.666010946315329, −0.9316440660451555,
0.9316440660451555, 1.666010946315329, 2.304538115017910, 3.001065602676283, 4.076892369769830, 4.923177176383010, 5.205106129123606, 6.202644862232076, 6.380427008777749, 7.174459875719406, 8.000636147893708, 8.656213814568994, 9.200042326416303, 9.728318050942609, 10.28321776132704, 10.77677745087536, 11.48037547950694, 12.09353700060961, 12.70199590982396, 13.34181344426789, 13.85198886460771, 14.20961902959630, 14.70391372941421, 15.29583676731756, 16.03728561903429
