| L(s) = 1 | + 2-s − 4-s + 5-s − 7-s − 3·8-s + 10-s − 2·11-s + 2·13-s − 14-s − 16-s − 2·19-s − 20-s − 2·22-s + 2·23-s + 25-s + 2·26-s + 28-s − 6·29-s + 8·31-s + 5·32-s − 35-s + 8·37-s − 2·38-s − 3·40-s − 2·41-s + 4·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s + 0.316·10-s − 0.603·11-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.426·22-s + 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.883·32-s − 0.169·35-s + 1.31·37-s − 0.324·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.93456773007996, −13.61634317115016, −13.10924740739405, −12.76496982016431, −12.44017746020014, −11.67572312988910, −11.18176020481746, −10.73392905203153, −9.949064257462915, −9.733406669018569, −9.137715224432515, −8.631277703930738, −8.122737866024210, −7.579876238044682, −6.794034863036204, −6.203243582999792, −5.963675683044863, −5.272807634951963, −4.815340488626456, −4.203759597534308, −3.716943427723428, −2.909333137319260, −2.666212668133798, −1.691025469242836, −0.8429337344858247, 0,
0.8429337344858247, 1.691025469242836, 2.666212668133798, 2.909333137319260, 3.716943427723428, 4.203759597534308, 4.815340488626456, 5.272807634951963, 5.963675683044863, 6.203243582999792, 6.794034863036204, 7.579876238044682, 8.122737866024210, 8.631277703930738, 9.137715224432515, 9.733406669018569, 9.949064257462915, 10.73392905203153, 11.18176020481746, 11.67572312988910, 12.44017746020014, 12.76496982016431, 13.10924740739405, 13.61634317115016, 13.93456773007996
