| L(s) = 1 | + 3.18·3-s + 3.43·5-s − 7-s + 7.16·9-s + 2.57·11-s − 1.58·13-s + 10.9·15-s − 4.77·17-s − 8.65·19-s − 3.18·21-s − 23-s + 6.80·25-s + 13.2·27-s + 7.67·29-s + 5.02·31-s + 8.21·33-s − 3.43·35-s + 3.71·37-s − 5.06·39-s + 11.9·41-s − 7.46·43-s + 24.6·45-s + 4.43·47-s + 49-s − 15.2·51-s + 8.06·53-s + 8.84·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s + 1.53·5-s − 0.377·7-s + 2.38·9-s + 0.776·11-s − 0.440·13-s + 2.82·15-s − 1.15·17-s − 1.98·19-s − 0.695·21-s − 0.208·23-s + 1.36·25-s + 2.55·27-s + 1.42·29-s + 0.902·31-s + 1.42·33-s − 0.580·35-s + 0.611·37-s − 0.811·39-s + 1.86·41-s − 1.13·43-s + 3.66·45-s + 0.647·47-s + 0.142·49-s − 2.13·51-s + 1.10·53-s + 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.584083450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.584083450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 8.65T + 19T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.855066204570564878122100150713, −8.528535280064286356706762036175, −7.41135772422518999947988523202, −6.51935512183333232660631041229, −6.16550251919339073979392063949, −4.58336937810814645699455991768, −4.10267004265373641375002956016, −2.69693843966312030414309030497, −2.41911767784779464661408532241, −1.46303753256391603658564211368,
1.46303753256391603658564211368, 2.41911767784779464661408532241, 2.69693843966312030414309030497, 4.10267004265373641375002956016, 4.58336937810814645699455991768, 6.16550251919339073979392063949, 6.51935512183333232660631041229, 7.41135772422518999947988523202, 8.528535280064286356706762036175, 8.855066204570564878122100150713
