| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 5·11-s − 12-s + 13-s − 15-s + 16-s + 17-s + 18-s + 20-s + 5·22-s − 6·23-s − 24-s + 25-s + 26-s − 27-s − 9·29-s − 30-s − 7·31-s + 32-s − 5·33-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 + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.223·20-s + 1.06·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.67·29-s − 0.182·30-s − 1.25·31-s + 0.176·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.740775884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.740775884\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.24343014680024, −13.06275134008013, −12.42712930494466, −11.93987251189446, −11.53327325724588, −11.26939924149857, −10.52558940124828, −10.20885177106571, −9.508212862421853, −9.177621198069145, −8.644548133789006, −7.860565061559676, −7.353947617011322, −6.873221069419249, −6.336357660915201, −5.874780127202071, −5.559529936347418, −4.944321604349667, −4.224708766608700, −3.752653281021976, −3.508613722492007, −2.455004213097535, −1.806353249062459, −1.448525918649032, −0.5179144180649840,
0.5179144180649840, 1.448525918649032, 1.806353249062459, 2.455004213097535, 3.508613722492007, 3.752653281021976, 4.224708766608700, 4.944321604349667, 5.559529936347418, 5.874780127202071, 6.336357660915201, 6.873221069419249, 7.353947617011322, 7.860565061559676, 8.644548133789006, 9.177621198069145, 9.508212862421853, 10.20885177106571, 10.52558940124828, 11.26939924149857, 11.53327325724588, 11.93987251189446, 12.42712930494466, 13.06275134008013, 13.24343014680024
