| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s + 4·11-s − 4·14-s + 16-s − 3·17-s − 20-s + 4·22-s + 4·23-s − 4·25-s − 4·28-s + 29-s − 4·31-s + 32-s − 3·34-s + 4·35-s − 3·37-s − 40-s − 9·41-s − 8·43-s + 4·44-s + 4·46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.223·20-s + 0.852·22-s + 0.834·23-s − 4/5·25-s − 0.755·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.676·35-s − 0.493·37-s − 0.158·40-s − 1.40·41-s − 1.21·43-s + 0.603·44-s + 0.589·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.428859473808084207561020932404, −7.19606273654421405538885269396, −6.77524713880413823780262477548, −6.18618951960893685878033201112, −5.25677768350654311306231689826, −4.22474183019187369465714215184, −3.59661605898965862469837055196, −2.94218201436595968970244853289, −1.63353613559349910686107745515, 0,
1.63353613559349910686107745515, 2.94218201436595968970244853289, 3.59661605898965862469837055196, 4.22474183019187369465714215184, 5.25677768350654311306231689826, 6.18618951960893685878033201112, 6.77524713880413823780262477548, 7.19606273654421405538885269396, 8.428859473808084207561020932404
