| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s − 17-s − 4·23-s − 5·25-s + 2·26-s + 28-s + 4·29-s + 32-s − 34-s + 8·37-s − 2·41-s − 4·46-s + 49-s − 5·50-s + 2·52-s − 2·53-s + 56-s + 4·58-s − 4·59-s + 12·61-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.834·23-s − 25-s + 0.392·26-s + 0.188·28-s + 0.742·29-s + 0.176·32-s − 0.171·34-s + 1.31·37-s − 0.312·41-s − 0.589·46-s + 1/7·49-s − 0.707·50-s + 0.277·52-s − 0.274·53-s + 0.133·56-s + 0.525·58-s − 0.520·59-s + 1.53·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.08837704730578, −12.64891495173572, −12.08142314591857, −11.69529079843828, −11.34603025705638, −10.88021937234617, −10.28213544109268, −9.968706811993166, −9.379998034371593, −8.776574360072730, −8.339496406117831, −7.744939308388009, −7.560524942488214, −6.711243717942047, −6.377452474798230, −5.870428055756084, −5.459396878022407, −4.793665764842031, −4.346835464007454, −3.902307978976641, −3.374449765075807, −2.677679390199266, −2.193344861007775, −1.566506619000954, −0.9387447957501033, 0,
0.9387447957501033, 1.566506619000954, 2.193344861007775, 2.677679390199266, 3.374449765075807, 3.902307978976641, 4.346835464007454, 4.793665764842031, 5.459396878022407, 5.870428055756084, 6.377452474798230, 6.711243717942047, 7.560524942488214, 7.744939308388009, 8.339496406117831, 8.776574360072730, 9.379998034371593, 9.968706811993166, 10.28213544109268, 10.88021937234617, 11.34603025705638, 11.69529079843828, 12.08142314591857, 12.64891495173572, 13.08837704730578
