| L(s) = 1 | + 5-s − 7-s − 5·13-s − 4·17-s + 3·19-s + 25-s − 7·29-s − 7·31-s − 35-s − 8·37-s − 41-s + 43-s − 6·47-s − 6·49-s + 2·53-s + 5·61-s − 5·65-s − 11·67-s + 6·71-s − 5·73-s − 79-s − 8·83-s − 4·85-s + 12·89-s + 5·91-s + 3·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.38·13-s − 0.970·17-s + 0.688·19-s + 1/5·25-s − 1.29·29-s − 1.25·31-s − 0.169·35-s − 1.31·37-s − 0.156·41-s + 0.152·43-s − 0.875·47-s − 6/7·49-s + 0.274·53-s + 0.640·61-s − 0.620·65-s − 1.34·67-s + 0.712·71-s − 0.585·73-s − 0.112·79-s − 0.878·83-s − 0.433·85-s + 1.27·89-s + 0.524·91-s + 0.307·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9569143530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9569143530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.97553777016138, −14.59270305831251, −14.11550102847840, −13.43708993425826, −12.90295626148971, −12.68697734961176, −11.75507319200859, −11.55268434130141, −10.71534361410822, −10.24007489701872, −9.689650088673883, −9.140634996741597, −8.842274352641480, −7.848869357447233, −7.378665644293186, −6.868855838165052, −6.284013691653720, −5.462392434734084, −5.126233022442936, −4.408042059182416, −3.580383905410117, −3.001911545728620, −2.129993035330794, −1.694125595093256, −0.3502303500500625,
0.3502303500500625, 1.694125595093256, 2.129993035330794, 3.001911545728620, 3.580383905410117, 4.408042059182416, 5.126233022442936, 5.462392434734084, 6.284013691653720, 6.868855838165052, 7.378665644293186, 7.848869357447233, 8.842274352641480, 9.140634996741597, 9.689650088673883, 10.24007489701872, 10.71534361410822, 11.55268434130141, 11.75507319200859, 12.68697734961176, 12.90295626148971, 13.43708993425826, 14.11550102847840, 14.59270305831251, 14.97553777016138
