| L(s) = 1 | − 3·3-s − 3·5-s − 7-s + 6·9-s + 4·11-s + 9·15-s + 2·17-s − 4·19-s + 3·21-s − 23-s + 4·25-s − 9·27-s + 2·29-s − 10·31-s − 12·33-s + 3·35-s + 4·37-s − 12·41-s − 18·45-s + 10·47-s + 49-s − 6·51-s + 4·53-s − 12·55-s + 12·57-s + 9·59-s − 13·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 0.377·7-s + 2·9-s + 1.20·11-s + 2.32·15-s + 0.485·17-s − 0.917·19-s + 0.654·21-s − 0.208·23-s + 4/5·25-s − 1.73·27-s + 0.371·29-s − 1.79·31-s − 2.08·33-s + 0.507·35-s + 0.657·37-s − 1.87·41-s − 2.68·45-s + 1.45·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s − 1.61·55-s + 1.58·57-s + 1.17·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.34029363604590, −13.81545926570145, −12.94074524625568, −12.62463591094514, −12.16402472511397, −11.79712586495899, −11.42281308333115, −10.93117182582454, −10.50779989840781, −9.935874329395205, −9.326642628419047, −8.710596309974901, −8.136003130107898, −7.435933040957177, −6.916018926463807, −6.698348115990298, −5.905478758790913, −5.611471563716348, −4.775003846070935, −4.353752038398124, −3.735136261577680, −3.463845457162694, −2.204839808374475, −1.356003063809473, −0.6428574269167663, 0,
0.6428574269167663, 1.356003063809473, 2.204839808374475, 3.463845457162694, 3.735136261577680, 4.353752038398124, 4.775003846070935, 5.611471563716348, 5.905478758790913, 6.698348115990298, 6.916018926463807, 7.435933040957177, 8.136003130107898, 8.710596309974901, 9.326642628419047, 9.935874329395205, 10.50779989840781, 10.93117182582454, 11.42281308333115, 11.79712586495899, 12.16402472511397, 12.62463591094514, 12.94074524625568, 13.81545926570145, 14.34029363604590
