| L(s) = 1 | + 3-s + 2.45·5-s + 0.345·7-s + 9-s + 3.27·11-s + 0.782·13-s + 2.45·15-s + 4.36·17-s + 2.85·19-s + 0.345·21-s − 23-s + 1.02·25-s + 27-s + 29-s − 3.97·31-s + 3.27·33-s + 0.847·35-s + 11.6·37-s + 0.782·39-s + 9.72·41-s − 8.57·43-s + 2.45·45-s + 2.54·47-s − 6.88·49-s + 4.36·51-s − 5.89·53-s + 8.04·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.09·5-s + 0.130·7-s + 0.333·9-s + 0.988·11-s + 0.217·13-s + 0.633·15-s + 1.05·17-s + 0.655·19-s + 0.0753·21-s − 0.208·23-s + 0.205·25-s + 0.192·27-s + 0.185·29-s − 0.713·31-s + 0.570·33-s + 0.143·35-s + 1.92·37-s + 0.125·39-s + 1.51·41-s − 1.30·43-s + 0.365·45-s + 0.370·47-s − 0.982·49-s + 0.611·51-s − 0.809·53-s + 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.031650176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.031650176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 - 0.345T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 0.782T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 9.72T + 41T^{2} \) |
| 43 | \( 1 + 8.57T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 + 4.40T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + 9.99T + 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 6.69T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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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
−7.81443201628927931641008202829, −7.24410760233141081253346401955, −6.27835999101510461433771953610, −5.92023171986488784059298805608, −5.08616492557994984122129512907, −4.20947719956655832039000355983, −3.43652538471732341897877634399, −2.65631989122736270403431917429, −1.72282816450602747904574611962, −1.07140280232030911545878916901,
1.07140280232030911545878916901, 1.72282816450602747904574611962, 2.65631989122736270403431917429, 3.43652538471732341897877634399, 4.20947719956655832039000355983, 5.08616492557994984122129512907, 5.92023171986488784059298805608, 6.27835999101510461433771953610, 7.24410760233141081253346401955, 7.81443201628927931641008202829
