L(s) = 1 | + 3·3-s − 5-s + 5·7-s + 6·9-s + 4·11-s − 13-s − 3·15-s − 3·17-s − 19-s + 15·21-s − 7·23-s + 25-s + 9·27-s − 3·29-s + 2·31-s + 12·33-s − 5·35-s − 2·37-s − 3·39-s − 6·41-s − 6·43-s − 6·45-s + 18·49-s − 9·51-s − 13·53-s − 4·55-s − 3·57-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s + 1.88·7-s + 2·9-s + 1.20·11-s − 0.277·13-s − 0.774·15-s − 0.727·17-s − 0.229·19-s + 3.27·21-s − 1.45·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s + 0.359·31-s + 2.08·33-s − 0.845·35-s − 0.328·37-s − 0.480·39-s − 0.937·41-s − 0.914·43-s − 0.894·45-s + 18/7·49-s − 1.26·51-s − 1.78·53-s − 0.539·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.616726418\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.616726418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.183429503828050666940148576109, −8.520988003692393874722984195719, −8.061666739050806025469758620561, −7.44216005547409771051517763477, −6.47811793943936770545207122255, −4.92569424984550777271351024658, −4.23023254095495088733882440215, −3.53749920190274767608809877312, −2.16076140731255439085089799617, −1.58735831285927584358770364342,
1.58735831285927584358770364342, 2.16076140731255439085089799617, 3.53749920190274767608809877312, 4.23023254095495088733882440215, 4.92569424984550777271351024658, 6.47811793943936770545207122255, 7.44216005547409771051517763477, 8.061666739050806025469758620561, 8.520988003692393874722984195719, 9.183429503828050666940148576109