L(s) = 1 | + 5-s + 7-s − 4·11-s + 4·13-s + 6·17-s − 4·19-s + 23-s + 25-s − 4·29-s + 8·31-s + 35-s + 2·37-s + 2·41-s + 49-s − 8·53-s − 4·55-s − 10·59-s − 8·61-s + 4·65-s + 4·67-s + 8·71-s − 4·77-s − 6·79-s − 6·83-s + 6·85-s + 14·89-s + 4·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.742·29-s + 1.43·31-s + 0.169·35-s + 0.328·37-s + 0.312·41-s + 1/7·49-s − 1.09·53-s − 0.539·55-s − 1.30·59-s − 1.02·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 0.455·77-s − 0.675·79-s − 0.658·83-s + 0.650·85-s + 1.48·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.97983865291270, −13.34243388443417, −12.87351027059653, −12.52486998332888, −11.95239647804729, −11.28098101741750, −10.91323122742555, −10.42989576739275, −10.07536948722346, −9.426976113518724, −8.970769083938215, −8.278861369517176, −7.901222530063345, −7.658850210765258, −6.709842511866772, −6.313016639657957, −5.691807928533729, −5.380081838301260, −4.682734918379023, −4.190542950209872, −3.387848846986339, −2.923316622784501, −2.286448396588790, −1.524321572448875, −1.002868621252759, 0,
1.002868621252759, 1.524321572448875, 2.286448396588790, 2.923316622784501, 3.387848846986339, 4.190542950209872, 4.682734918379023, 5.380081838301260, 5.691807928533729, 6.313016639657957, 6.709842511866772, 7.658850210765258, 7.901222530063345, 8.278861369517176, 8.970769083938215, 9.426976113518724, 10.07536948722346, 10.42989576739275, 10.91323122742555, 11.28098101741750, 11.95239647804729, 12.52486998332888, 12.87351027059653, 13.34243388443417, 13.97983865291270