| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s + 14-s − 2·15-s + 16-s + 3·17-s − 18-s + 2·20-s + 21-s − 4·22-s − 23-s + 24-s − 25-s + 2·26-s − 27-s − 28-s + 5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.48071865140230, −12.26861875978793, −11.86287847222526, −11.49065997363897, −10.84797108232449, −10.26922527745100, −10.15458268678043, −9.615073880244893, −9.195228570849387, −8.924542653331854, −8.149887788292247, −7.776497882411277, −7.157319286125370, −6.606888227586180, −6.441848674107880, −5.894868779359381, −5.405737521501123, −4.903072419320393, −4.298618850450715, −3.556779633857025, −3.236746088159983, −2.329343227191453, −1.972066753403961, −1.285215439648488, −0.8212340938540528, 0,
0.8212340938540528, 1.285215439648488, 1.972066753403961, 2.329343227191453, 3.236746088159983, 3.556779633857025, 4.298618850450715, 4.903072419320393, 5.405737521501123, 5.894868779359381, 6.441848674107880, 6.606888227586180, 7.157319286125370, 7.776497882411277, 8.149887788292247, 8.924542653331854, 9.195228570849387, 9.615073880244893, 10.15458268678043, 10.26922527745100, 10.84797108232449, 11.49065997363897, 11.86287847222526, 12.26861875978793, 12.48071865140230
