| L(s) = 1 | − 5-s − 11-s + 13-s + 3·17-s + 19-s − 2·23-s + 25-s + 2·31-s + 37-s + 7·41-s − 13·43-s − 9·47-s − 7·49-s − 4·53-s + 55-s − 14·59-s − 10·61-s − 65-s + 67-s − 10·71-s − 4·73-s − 2·79-s + 2·83-s − 3·85-s + 14·89-s − 95-s − 17·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.277·13-s + 0.727·17-s + 0.229·19-s − 0.417·23-s + 1/5·25-s + 0.359·31-s + 0.164·37-s + 1.09·41-s − 1.98·43-s − 1.31·47-s − 49-s − 0.549·53-s + 0.134·55-s − 1.82·59-s − 1.28·61-s − 0.124·65-s + 0.122·67-s − 1.18·71-s − 0.468·73-s − 0.225·79-s + 0.219·83-s − 0.325·85-s + 1.48·89-s − 0.102·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208139444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208139444\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.80894571587351, −13.13881355271059, −12.81322373786781, −12.20673925918904, −11.75173935522559, −11.39284504462437, −10.74366530162667, −10.35554516486536, −9.741622073991766, −9.369811909397835, −8.703139176944131, −8.087390928583661, −7.851891179982852, −7.310115228046393, −6.594062688924325, −6.149021806825470, −5.629011402180263, −4.789924220784785, −4.643617698154954, −3.760133960262966, −3.196389716521078, −2.852592431243466, −1.797283958426174, −1.348659363449490, −0.3470567926265427,
0.3470567926265427, 1.348659363449490, 1.797283958426174, 2.852592431243466, 3.196389716521078, 3.760133960262966, 4.643617698154954, 4.789924220784785, 5.629011402180263, 6.149021806825470, 6.594062688924325, 7.310115228046393, 7.851891179982852, 8.087390928583661, 8.703139176944131, 9.369811909397835, 9.741622073991766, 10.35554516486536, 10.74366530162667, 11.39284504462437, 11.75173935522559, 12.20673925918904, 12.81322373786781, 13.13881355271059, 13.80894571587351
