| L(s) = 1 | + 15.8·5-s − 7·7-s − 15.8·11-s + 26·13-s − 79.3·17-s − 68·19-s − 47.6·23-s + 127.·25-s − 253.·29-s − 212·31-s − 111.·35-s + 218·37-s + 396.·41-s − 260·43-s − 412.·47-s + 49·49-s − 476.·53-s − 252.·55-s + 285.·59-s − 322·61-s + 412.·65-s − 356·67-s + 1.12e3·71-s − 226·73-s + 111.·77-s − 440·79-s + 253.·83-s + ⋯ |
| L(s) = 1 | + 1.41·5-s − 0.377·7-s − 0.435·11-s + 0.554·13-s − 1.13·17-s − 0.821·19-s − 0.431·23-s + 1.01·25-s − 1.62·29-s − 1.22·31-s − 0.536·35-s + 0.968·37-s + 1.51·41-s − 0.922·43-s − 1.28·47-s + 0.142·49-s − 1.23·53-s − 0.617·55-s + 0.630·59-s − 0.675·61-s + 0.787·65-s − 0.649·67-s + 1.88·71-s − 0.362·73-s + 0.164·77-s − 0.626·79-s + 0.335·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 260T + 7.95e4T^{2} \) |
| 47 | \( 1 + 412.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 + 356T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 226T + 3.89e5T^{2} \) |
| 79 | \( 1 + 440T + 4.93e5T^{2} \) |
| 83 | \( 1 - 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−9.331211535745703880552968747178, −8.517796699894632042056981661086, −7.41893656686633331716498410984, −6.33000236044839673909579131745, −5.93778509100276828821968518098, −4.90202353521866779106903073803, −3.74006598792064741473100057929, −2.43851888460145800031931351781, −1.69509888059959239504999121409, 0,
1.69509888059959239504999121409, 2.43851888460145800031931351781, 3.74006598792064741473100057929, 4.90202353521866779106903073803, 5.93778509100276828821968518098, 6.33000236044839673909579131745, 7.41893656686633331716498410984, 8.517796699894632042056981661086, 9.331211535745703880552968747178
