L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s − 2·11-s + 12-s + 13-s − 16-s − 17-s − 18-s + 2·22-s − 8·23-s − 3·24-s − 5·25-s − 26-s − 27-s − 6·29-s − 6·31-s − 5·32-s + 2·33-s + 34-s − 36-s + 4·37-s − 39-s + 12·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.277·13-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.426·22-s − 1.66·23-s − 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.883·32-s + 0.348·33-s + 0.171·34-s − 1/6·36-s + 0.657·37-s − 0.160·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.54919374578202, −14.71395869803655, −14.25683847515861, −13.60707153330633, −13.21615307480334, −12.65675761933695, −12.18415644049152, −11.29864747058840, −11.07324401668395, −10.45353379404106, −9.850665530144944, −9.477260288807477, −8.961744391812376, −8.106577370337724, −7.796466131578672, −7.365477310021775, −6.421183103466588, −5.872431182239603, −5.355430213685897, −4.661441645687555, −3.986738823857492, −3.563027391399521, −2.242846755609036, −1.790511533655810, −0.7110840493957610, 0,
0.7110840493957610, 1.790511533655810, 2.242846755609036, 3.563027391399521, 3.986738823857492, 4.661441645687555, 5.355430213685897, 5.872431182239603, 6.421183103466588, 7.365477310021775, 7.796466131578672, 8.106577370337724, 8.961744391812376, 9.477260288807477, 9.850665530144944, 10.45353379404106, 11.07324401668395, 11.29864747058840, 12.18415644049152, 12.65675761933695, 13.21615307480334, 13.60707153330633, 14.25683847515861, 14.71395869803655, 15.54919374578202