| L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 2·13-s + 6·17-s + 4·21-s − 23-s − 27-s − 4·31-s + 33-s + 2·37-s + 2·39-s + 4·43-s + 9·49-s − 6·51-s − 4·53-s − 14·59-s − 4·63-s − 4·67-s + 69-s + 10·71-s + 6·73-s + 4·77-s + 8·79-s + 81-s − 10·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.872·21-s − 0.208·23-s − 0.192·27-s − 0.718·31-s + 0.174·33-s + 0.328·37-s + 0.320·39-s + 0.609·43-s + 9/7·49-s − 0.840·51-s − 0.549·53-s − 1.82·59-s − 0.503·63-s − 0.488·67-s + 0.120·69-s + 1.18·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-s + 1/9·81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.68131547430472, −12.52953700583079, −12.10630896415878, −11.66385595006667, −10.94341243034972, −10.62956749436060, −10.12835684554751, −9.713892582919205, −9.361235031724492, −9.011918659019872, −8.037829326925876, −7.838960448383250, −7.237542819279751, −6.822682689664400, −6.215537865963528, −5.958551491836331, −5.358319962278857, −4.990899250452906, −4.247002503516673, −3.724574217558585, −3.194669115199191, −2.798995633649088, −2.081461358078283, −1.317141163403778, −0.6021481880260732, 0,
0.6021481880260732, 1.317141163403778, 2.081461358078283, 2.798995633649088, 3.194669115199191, 3.724574217558585, 4.247002503516673, 4.990899250452906, 5.358319962278857, 5.958551491836331, 6.215537865963528, 6.822682689664400, 7.237542819279751, 7.838960448383250, 8.037829326925876, 9.011918659019872, 9.361235031724492, 9.713892582919205, 10.12835684554751, 10.62956749436060, 10.94341243034972, 11.66385595006667, 12.10630896415878, 12.52953700583079, 12.68131547430472
