| L(s) = 1 | + 3·3-s + 6·5-s − 4·7-s + 9·9-s − 36·11-s − 13·13-s + 18·15-s + 66·17-s − 56·19-s − 12·21-s + 96·23-s − 89·25-s + 27·27-s − 222·29-s + 260·31-s − 108·33-s − 24·35-s + 106·37-s − 39·39-s − 90·41-s − 44·43-s + 54·45-s + 168·47-s − 327·49-s + 198·51-s − 30·53-s − 216·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.536·5-s − 0.215·7-s + 1/3·9-s − 0.986·11-s − 0.277·13-s + 0.309·15-s + 0.941·17-s − 0.676·19-s − 0.124·21-s + 0.870·23-s − 0.711·25-s + 0.192·27-s − 1.42·29-s + 1.50·31-s − 0.569·33-s − 0.115·35-s + 0.470·37-s − 0.160·39-s − 0.342·41-s − 0.156·43-s + 0.178·45-s + 0.521·47-s − 0.953·49-s + 0.543·51-s − 0.0777·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 222 T + p^{3} T^{2} \) |
| 31 | \( 1 - 260 T + p^{3} T^{2} \) |
| 37 | \( 1 - 106 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 44 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 30 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 346 T + p^{3} T^{2} \) |
| 67 | \( 1 - 256 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 814 T + p^{3} T^{2} \) |
| 79 | \( 1 - 200 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 318 T + p^{3} T^{2} \) |
| 97 | \( 1 + 502 T + p^{3} T^{2} \) |
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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
−8.112698940536425295509715462841, −7.59679234798732896964054767787, −6.69209932549194183501725531014, −5.80181239501712160878120488494, −5.10467449531652369636217821988, −4.11940166909757614381779320694, −3.08245596103226979762529896402, −2.41295486777093459174346705919, −1.37346545128720989576313108927, 0,
1.37346545128720989576313108927, 2.41295486777093459174346705919, 3.08245596103226979762529896402, 4.11940166909757614381779320694, 5.10467449531652369636217821988, 5.80181239501712160878120488494, 6.69209932549194183501725531014, 7.59679234798732896964054767787, 8.112698940536425295509715462841
