| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 12-s + 4·13-s + 15-s + 16-s − 17-s + 18-s + 6·19-s − 20-s − 22-s − 2·23-s − 24-s + 25-s + 4·26-s − 27-s − 6·29-s + 30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.213·22-s − 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.90981982974539, −12.47222762450521, −12.14229538112604, −11.64339227022120, −11.18069440418725, −10.87393247626852, −10.49034406487331, −9.850690929202154, −9.239963714125281, −8.972400666384987, −8.077183553145327, −7.825730482643082, −7.327410494520224, −6.827976646470585, −6.228407750602259, −5.858204128111074, −5.361268826965771, −4.981493453411430, −4.176880307914388, −3.953220428883178, −3.414090391547117, −2.734252366465018, −2.212522229985104, −1.296546094547709, −0.9357644787509199, 0,
0.9357644787509199, 1.296546094547709, 2.212522229985104, 2.734252366465018, 3.414090391547117, 3.953220428883178, 4.176880307914388, 4.981493453411430, 5.361268826965771, 5.858204128111074, 6.228407750602259, 6.827976646470585, 7.327410494520224, 7.825730482643082, 8.077183553145327, 8.972400666384987, 9.239963714125281, 9.850690929202154, 10.49034406487331, 10.87393247626852, 11.18069440418725, 11.64339227022120, 12.14229538112604, 12.47222762450521, 12.90981982974539
