| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3·7-s + 8-s + 9-s + 11-s − 12-s + 2·13-s − 3·14-s + 16-s − 7·17-s + 18-s − 7·19-s + 3·21-s + 22-s + 3·23-s − 24-s − 5·25-s + 2·26-s − 27-s − 3·28-s + 2·29-s + 4·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 1.60·19-s + 0.654·21-s + 0.213·22-s + 0.625·23-s − 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s − 0.566·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.82056364236173, −13.51461291305839, −13.13311894555231, −12.77370123951124, −12.12182682439804, −11.80051091714719, −11.10836565038350, −10.81465295054220, −10.25525252109784, −9.784161474375953, −9.088400116452228, −8.620786722887606, −8.153606611830001, −7.153655301782985, −6.754285902104749, −6.494428322677085, −5.964676531272570, −5.467554215148236, −4.634173230432061, −4.212402156847804, −3.800590692871942, −3.050615888658549, −2.348922887416412, −1.840527295817601, −0.7782516708930054, 0,
0.7782516708930054, 1.840527295817601, 2.348922887416412, 3.050615888658549, 3.800590692871942, 4.212402156847804, 4.634173230432061, 5.467554215148236, 5.964676531272570, 6.494428322677085, 6.754285902104749, 7.153655301782985, 8.153606611830001, 8.620786722887606, 9.088400116452228, 9.784161474375953, 10.25525252109784, 10.81465295054220, 11.10836565038350, 11.80051091714719, 12.12182682439804, 12.77370123951124, 13.13311894555231, 13.51461291305839, 13.82056364236173
