| L(s) = 1 | + 3-s − 2.98·5-s + 2.49·7-s + 9-s − 1.94·11-s − 3.49·13-s − 2.98·15-s + 4.53·17-s − 2.24·19-s + 2.49·21-s − 1.85·23-s + 3.88·25-s + 27-s + 7.87·29-s + 3.36·31-s − 1.94·33-s − 7.42·35-s − 6.58·37-s − 3.49·39-s + 1.44·41-s − 11.7·43-s − 2.98·45-s + 5.12·47-s − 0.788·49-s + 4.53·51-s + 3.51·53-s + 5.79·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.33·5-s + 0.942·7-s + 0.333·9-s − 0.586·11-s − 0.968·13-s − 0.769·15-s + 1.10·17-s − 0.515·19-s + 0.543·21-s − 0.386·23-s + 0.776·25-s + 0.192·27-s + 1.46·29-s + 0.604·31-s − 0.338·33-s − 1.25·35-s − 1.08·37-s − 0.559·39-s + 0.225·41-s − 1.78·43-s − 0.444·45-s + 0.747·47-s − 0.112·49-s + 0.635·51-s + 0.482·53-s + 0.781·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.59T + 61T^{2} \) |
| 67 | \( 1 + 0.605T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 - 0.536T + 79T^{2} \) |
| 83 | \( 1 + 2.30T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73808013889184202570574068400, −7.36753262969791574731179803244, −6.48436516326379569964416103908, −5.27571727784251809589396798223, −4.75043559951084547773915938290, −4.06034633842076479732640500127, −3.21860185582497083783870595786, −2.46104106370571321706107309592, −1.33490575494621993760299987872, 0,
1.33490575494621993760299987872, 2.46104106370571321706107309592, 3.21860185582497083783870595786, 4.06034633842076479732640500127, 4.75043559951084547773915938290, 5.27571727784251809589396798223, 6.48436516326379569964416103908, 7.36753262969791574731179803244, 7.73808013889184202570574068400
