| L(s) = 1 | + 3-s + 26·7-s − 26·9-s + 45·11-s + 44·13-s + 117·17-s − 91·19-s + 26·21-s − 18·23-s − 53·27-s + 144·29-s + 26·31-s + 45·33-s − 214·37-s + 44·39-s − 459·41-s − 460·43-s − 468·47-s + 333·49-s + 117·51-s + 558·53-s − 91·57-s − 72·59-s − 118·61-s − 676·63-s + 251·67-s − 18·69-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 1.40·7-s − 0.962·9-s + 1.23·11-s + 0.938·13-s + 1.66·17-s − 1.09·19-s + 0.270·21-s − 0.163·23-s − 0.377·27-s + 0.922·29-s + 0.150·31-s + 0.237·33-s − 0.950·37-s + 0.180·39-s − 1.74·41-s − 1.63·43-s − 1.45·47-s + 0.970·49-s + 0.321·51-s + 1.44·53-s − 0.211·57-s − 0.158·59-s − 0.247·61-s − 1.35·63-s + 0.457·67-s − 0.0314·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.897615980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897615980\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 45 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 117 T + p^{3} T^{2} \) |
| 19 | \( 1 + 91 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 26 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 459 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 + 468 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 72 T + p^{3} T^{2} \) |
| 61 | \( 1 + 118 T + p^{3} T^{2} \) |
| 67 | \( 1 - 251 T + p^{3} T^{2} \) |
| 71 | \( 1 - 108 T + p^{3} T^{2} \) |
| 73 | \( 1 - 299 T + p^{3} T^{2} \) |
| 79 | \( 1 + 898 T + p^{3} T^{2} \) |
| 83 | \( 1 - 927 T + p^{3} T^{2} \) |
| 89 | \( 1 - 351 T + p^{3} T^{2} \) |
| 97 | \( 1 - 386 T + p^{3} T^{2} \) |
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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
−13.69423873277935682711355843997, −12.02636464936227526894956006362, −11.46016321284638139365905602361, −10.26003400696340129781445676078, −8.671120000818276612817751503417, −8.170182302306989191934356478754, −6.47911250905803080175356251086, −5.14413069161983928991275934925, −3.58348405288454597633490267851, −1.52401419890921695591063204835,
1.52401419890921695591063204835, 3.58348405288454597633490267851, 5.14413069161983928991275934925, 6.47911250905803080175356251086, 8.170182302306989191934356478754, 8.671120000818276612817751503417, 10.26003400696340129781445676078, 11.46016321284638139365905602361, 12.02636464936227526894956006362, 13.69423873277935682711355843997
