| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s + 4·11-s − 5·13-s + 2·15-s + 5·17-s + 7·19-s − 2·21-s − 23-s − 25-s − 27-s − 29-s + 8·31-s − 4·33-s − 4·35-s − 3·37-s + 5·39-s + 4·41-s + 7·43-s − 2·45-s − 8·47-s − 3·49-s − 5·51-s + 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.38·13-s + 0.516·15-s + 1.21·17-s + 1.60·19-s − 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 1.43·31-s − 0.696·33-s − 0.676·35-s − 0.493·37-s + 0.800·39-s + 0.624·41-s + 1.06·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.700·51-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.650460783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650460783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.75381831022807808489865414675, −7.26298703541126069336965263942, −6.55596593307623273821455397648, −5.58065758948784366744089798944, −5.07052167628474645912791813683, −4.31757529480503423230685312436, −3.67927956227396799162209385805, −2.75143375418139018716919193655, −1.50979865351569568790482713476, −0.71143736655754546648035729554,
0.71143736655754546648035729554, 1.50979865351569568790482713476, 2.75143375418139018716919193655, 3.67927956227396799162209385805, 4.31757529480503423230685312436, 5.07052167628474645912791813683, 5.58065758948784366744089798944, 6.55596593307623273821455397648, 7.26298703541126069336965263942, 7.75381831022807808489865414675
