| L(s) = 1 | − 3·3-s − 5·5-s + 28·7-s + 9·9-s + 64·11-s + 2·13-s + 15·15-s − 74·17-s − 19·19-s − 84·21-s − 72·23-s + 25·25-s − 27·27-s − 310·29-s + 248·31-s − 192·33-s − 140·35-s − 158·37-s − 6·39-s − 462·41-s + 36·43-s − 45·45-s − 168·47-s + 441·49-s + 222·51-s + 82·53-s − 320·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.75·11-s + 0.0426·13-s + 0.258·15-s − 1.05·17-s − 0.229·19-s − 0.872·21-s − 0.652·23-s + 1/5·25-s − 0.192·27-s − 1.98·29-s + 1.43·31-s − 1.01·33-s − 0.676·35-s − 0.702·37-s − 0.0246·39-s − 1.75·41-s + 0.127·43-s − 0.149·45-s − 0.521·47-s + 9/7·49-s + 0.609·51-s + 0.212·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 64 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 310 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 36 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 82 T + p^{3} T^{2} \) |
| 59 | \( 1 + 504 T + p^{3} T^{2} \) |
| 61 | \( 1 + 706 T + p^{3} T^{2} \) |
| 67 | \( 1 - 844 T + p^{3} T^{2} \) |
| 71 | \( 1 - 224 T + p^{3} T^{2} \) |
| 73 | \( 1 + 894 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 - 892 T + p^{3} T^{2} \) |
| 89 | \( 1 - 266 T + p^{3} T^{2} \) |
| 97 | \( 1 + 822 T + p^{3} T^{2} \) |
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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
−8.348236633305258295639834679654, −7.46525846688906077010413401989, −6.71374631798623703083351838919, −5.98918130090157497639275749219, −4.91571406680102307944556280232, −4.35736904017840750949523629897, −3.61101668997510135742263801584, −1.95015318696105525846482014000, −1.33941425266061922740456195853, 0,
1.33941425266061922740456195853, 1.95015318696105525846482014000, 3.61101668997510135742263801584, 4.35736904017840750949523629897, 4.91571406680102307944556280232, 5.98918130090157497639275749219, 6.71374631798623703083351838919, 7.46525846688906077010413401989, 8.348236633305258295639834679654
