| L(s) = 1 | − 4·7-s − 4·11-s + 13-s − 2·17-s − 4·19-s − 2·23-s − 2·29-s + 6·31-s − 2·37-s − 2·41-s − 4·43-s − 12·47-s + 9·49-s − 8·59-s + 6·61-s + 10·67-s + 8·71-s − 4·73-s + 16·77-s + 4·79-s − 4·83-s + 2·89-s − 4·91-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 0.417·23-s − 0.371·29-s + 1.07·31-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 1.04·59-s + 0.768·61-s + 1.22·67-s + 0.949·71-s − 0.468·73-s + 1.82·77-s + 0.450·79-s − 0.439·83-s + 0.211·89-s − 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.94482655398547, −13.40221732111417, −13.07773247000005, −12.76135304688848, −12.25123606064565, −11.60807477665836, −11.05084985035503, −10.56004373197924, −9.938784603231879, −9.865638798040399, −9.122800958861733, −8.476879875984552, −8.193179414842022, −7.496041600645234, −6.868525272177504, −6.350252489062675, −6.139526388260417, −5.301022112178820, −4.829063103041340, −4.125883847565963, −3.462167390252457, −3.039422088626903, −2.368402570048391, −1.796411722430690, −0.6061171832596066, 0,
0.6061171832596066, 1.796411722430690, 2.368402570048391, 3.039422088626903, 3.462167390252457, 4.125883847565963, 4.829063103041340, 5.301022112178820, 6.139526388260417, 6.350252489062675, 6.868525272177504, 7.496041600645234, 8.193179414842022, 8.476879875984552, 9.122800958861733, 9.865638798040399, 9.938784603231879, 10.56004373197924, 11.05084985035503, 11.60807477665836, 12.25123606064565, 12.76135304688848, 13.07773247000005, 13.40221732111417, 13.94482655398547
