L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 8·17-s + 18-s − 19-s − 20-s − 2·21-s + 24-s − 4·25-s − 4·26-s + 27-s − 2·28-s − 6·29-s − 30-s − 9·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s + 0.204·24-s − 4/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.182·30-s − 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.24977320092718, −13.75301847663385, −13.11640552204659, −12.85951219303186, −12.58115696215251, −11.84702610543463, −11.29068187116155, −11.00081523480276, −10.38335864849867, −9.584580513885328, −9.407988172741980, −8.915290399137753, −8.069402991102701, −7.698343033519924, −7.220533756775885, −6.533147393217878, −6.353172562708772, −5.501671892970491, −4.825327558885636, −4.460236772414722, −3.774960913166492, −3.383599059724343, −2.706293158289772, −2.083752620191814, −1.654750477135382, 0, 0,
1.654750477135382, 2.083752620191814, 2.706293158289772, 3.383599059724343, 3.774960913166492, 4.460236772414722, 4.825327558885636, 5.501671892970491, 6.353172562708772, 6.533147393217878, 7.220533756775885, 7.698343033519924, 8.069402991102701, 8.915290399137753, 9.407988172741980, 9.584580513885328, 10.38335864849867, 11.00081523480276, 11.29068187116155, 11.84702610543463, 12.58115696215251, 12.85951219303186, 13.11640552204659, 13.75301847663385, 14.24977320092718