| L(s) = 1 | − 2·5-s − 2·7-s − 4·13-s + 17-s + 2·19-s + 2·23-s − 25-s + 2·29-s + 4·31-s + 4·35-s + 6·37-s + 6·41-s + 2·43-s − 3·49-s + 12·53-s − 14·59-s − 6·61-s + 8·65-s + 4·67-s − 2·71-s + 8·73-s − 2·79-s + 12·83-s − 2·85-s − 6·89-s + 8·91-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 1.10·13-s + 0.242·17-s + 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s − 3/7·49-s + 1.64·53-s − 1.82·59-s − 0.768·61-s + 0.992·65-s + 0.488·67-s − 0.237·71-s + 0.936·73-s − 0.225·79-s + 1.31·83-s − 0.216·85-s − 0.635·89-s + 0.838·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.62726367031550, −12.58073119232715, −12.01716283650650, −11.61648358226885, −11.22431199088221, −10.57052171600710, −10.20653258954551, −9.628369049935801, −9.345035453571072, −8.866141518337378, −8.103286575482107, −7.818727366834831, −7.383476652298107, −6.971331904283288, −6.276576977976099, −6.033463474941709, −5.181733351241165, −4.884248336252671, −4.184450338320616, −3.870140190800746, −3.137803977507665, −2.763373123114618, −2.254228389373139, −1.280319668836369, −0.6563597431732027, 0,
0.6563597431732027, 1.280319668836369, 2.254228389373139, 2.763373123114618, 3.137803977507665, 3.870140190800746, 4.184450338320616, 4.884248336252671, 5.181733351241165, 6.033463474941709, 6.276576977976099, 6.971331904283288, 7.383476652298107, 7.818727366834831, 8.103286575482107, 8.866141518337378, 9.345035453571072, 9.628369049935801, 10.20653258954551, 10.57052171600710, 11.22431199088221, 11.61648358226885, 12.01716283650650, 12.58073119232715, 12.62726367031550
