| L(s) = 1 | + 2-s + 2·3-s + 4-s + 4·5-s + 2·6-s + 7-s + 8-s + 9-s + 4·10-s − 2·11-s + 2·12-s + 2·13-s + 14-s + 8·15-s + 16-s + 7·17-s + 18-s + 4·20-s + 2·21-s − 2·22-s + 23-s + 2·24-s + 11·25-s + 2·26-s − 4·27-s + 28-s + 8·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s + 0.577·12-s + 0.554·13-s + 0.267·14-s + 2.06·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.894·20-s + 0.436·21-s − 0.426·22-s + 0.208·23-s + 0.408·24-s + 11/5·25-s + 0.392·26-s − 0.769·27-s + 0.188·28-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.01078938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.01078938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.79106363818340, −13.18611529441972, −12.83106917709787, −12.51479800044168, −11.68346188028553, −11.15113214819657, −10.54703863399058, −10.19613498047891, −9.645091382056464, −9.222915999660962, −8.750964156397550, −8.131639804438444, −7.665672600954054, −7.195073991547879, −6.424225937841360, −5.833834569471103, −5.571353119169153, −5.128629513744128, −4.356022049849561, −3.656718024489166, −3.058903996997183, −2.675803731114739, −2.085186481357756, −1.559154379893065, −0.9184463945435938,
0.9184463945435938, 1.559154379893065, 2.085186481357756, 2.675803731114739, 3.058903996997183, 3.656718024489166, 4.356022049849561, 5.128629513744128, 5.571353119169153, 5.833834569471103, 6.424225937841360, 7.195073991547879, 7.665672600954054, 8.131639804438444, 8.750964156397550, 9.222915999660962, 9.645091382056464, 10.19613498047891, 10.54703863399058, 11.15113214819657, 11.68346188028553, 12.51479800044168, 12.83106917709787, 13.18611529441972, 13.79106363818340
