| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 5·11-s − 12-s + 15-s + 16-s + 3·17-s − 18-s − 19-s − 20-s + 5·22-s + 3·23-s + 24-s − 4·25-s − 27-s + 9·29-s − 30-s + 4·31-s − 32-s + 5·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 1.06·22-s + 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + 0.870·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026716271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026716271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.63421627616061, −14.06293597578822, −13.34337416214382, −12.88466206558757, −12.44767753117065, −11.81116797328042, −11.35099274440617, −10.97043688756508, −10.29756009787391, −9.871563038595768, −9.621579832161268, −8.484321175102987, −8.240822559298565, −7.790827303723587, −7.215990861662073, −6.515496568099879, −6.128196480415211, −5.232021463854415, −4.997284440531726, −4.180604557680827, −3.374936753034452, −2.713114363533683, −2.129705153151397, −1.027578762334292, −0.5006991723387578,
0.5006991723387578, 1.027578762334292, 2.129705153151397, 2.713114363533683, 3.374936753034452, 4.180604557680827, 4.997284440531726, 5.232021463854415, 6.128196480415211, 6.515496568099879, 7.215990861662073, 7.790827303723587, 8.240822559298565, 8.484321175102987, 9.621579832161268, 9.871563038595768, 10.29756009787391, 10.97043688756508, 11.35099274440617, 11.81116797328042, 12.44767753117065, 12.88466206558757, 13.34337416214382, 14.06293597578822, 14.63421627616061
