| L(s) = 1 | + 5·3-s − 5·5-s + 7·7-s − 2·9-s + 15·11-s + 13·13-s − 25·15-s − 27·17-s + 154·19-s + 35·21-s − 186·23-s + 25·25-s − 145·27-s − 3·29-s − 328·31-s + 75·33-s − 35·35-s − 254·37-s + 65·39-s + 96·41-s − 134·43-s + 10·45-s + 51·47-s + 49·49-s − 135·51-s − 240·53-s − 75·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.447·5-s + 0.377·7-s − 0.0740·9-s + 0.411·11-s + 0.277·13-s − 0.430·15-s − 0.385·17-s + 1.85·19-s + 0.363·21-s − 1.68·23-s + 1/5·25-s − 1.03·27-s − 0.0192·29-s − 1.90·31-s + 0.395·33-s − 0.169·35-s − 1.12·37-s + 0.266·39-s + 0.365·41-s − 0.475·43-s + 0.0331·45-s + 0.158·47-s + 1/7·49-s − 0.370·51-s − 0.622·53-s − 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - p T + p^{3} T^{2} \) |
| 17 | \( 1 + 27 T + p^{3} T^{2} \) |
| 19 | \( 1 - 154 T + p^{3} T^{2} \) |
| 23 | \( 1 + 186 T + p^{3} T^{2} \) |
| 29 | \( 1 + 3 T + p^{3} T^{2} \) |
| 31 | \( 1 + 328 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 96 T + p^{3} T^{2} \) |
| 43 | \( 1 + 134 T + p^{3} T^{2} \) |
| 47 | \( 1 - 51 T + p^{3} T^{2} \) |
| 53 | \( 1 + 240 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 616 T + p^{3} T^{2} \) |
| 67 | \( 1 + 296 T + p^{3} T^{2} \) |
| 71 | \( 1 + 48 T + p^{3} T^{2} \) |
| 73 | \( 1 + 322 T + p^{3} T^{2} \) |
| 79 | \( 1 - 659 T + p^{3} T^{2} \) |
| 83 | \( 1 + 300 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 199 T + p^{3} T^{2} \) |
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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
−8.271772062696331412502534608083, −7.68313021324984011490048849094, −7.01143211414404987073319687842, −5.85829100338849782665213544504, −5.11451719239530930323722048859, −3.86248155076943027662634659765, −3.49892957502683401301698977279, −2.35800638675618619557069531774, −1.43485149136630018461025069912, 0,
1.43485149136630018461025069912, 2.35800638675618619557069531774, 3.49892957502683401301698977279, 3.86248155076943027662634659765, 5.11451719239530930323722048859, 5.85829100338849782665213544504, 7.01143211414404987073319687842, 7.68313021324984011490048849094, 8.271772062696331412502534608083
