| L(s) = 1 | − 2.41·3-s − 5-s + 2·7-s + 2.82·9-s + 0.414·11-s − 3.82·13-s + 2.41·15-s − 2.82·17-s + 3.65·19-s − 4.82·21-s + 6.82·23-s − 4·25-s + 0.414·27-s − 29-s − 0.414·31-s − 0.999·33-s − 2·35-s + 5.65·37-s + 9.24·39-s + 0.828·41-s − 5.24·43-s − 2.82·45-s + 1.58·47-s − 3·49-s + 6.82·51-s + 5.48·53-s − 0.414·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.447·5-s + 0.755·7-s + 0.942·9-s + 0.124·11-s − 1.06·13-s + 0.623·15-s − 0.685·17-s + 0.838·19-s − 1.05·21-s + 1.42·23-s − 0.800·25-s + 0.0797·27-s − 0.185·29-s − 0.0743·31-s − 0.174·33-s − 0.338·35-s + 0.929·37-s + 1.48·39-s + 0.129·41-s − 0.799·43-s − 0.421·45-s + 0.231·47-s − 0.428·49-s + 0.956·51-s + 0.753·53-s − 0.0558·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 31 | \( 1 + 0.414T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 8.89T + 79T^{2} \) |
| 83 | \( 1 - 4.48T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.901960975109664795419360457665, −7.84034813238954427733937353300, −7.22502405939388722231513331229, −6.44257538556520353211922172701, −5.44057229537643532192526269848, −4.92371286068695655605453713601, −4.16986857321832827512211503839, −2.76786087527848586964829931745, −1.32852997150164032090909125122, 0,
1.32852997150164032090909125122, 2.76786087527848586964829931745, 4.16986857321832827512211503839, 4.92371286068695655605453713601, 5.44057229537643532192526269848, 6.44257538556520353211922172701, 7.22502405939388722231513331229, 7.84034813238954427733937353300, 8.901960975109664795419360457665
