| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s + 2·11-s + 13-s + 16-s + 7·17-s + 3·18-s − 19-s − 20-s − 2·22-s + 6·23-s + 25-s − 26-s + 29-s − 3·31-s − 32-s − 7·34-s − 3·36-s + 4·37-s + 38-s + 40-s + 9·41-s − 43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s + 0.277·13-s + 1/4·16-s + 1.69·17-s + 0.707·18-s − 0.229·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.185·29-s − 0.538·31-s − 0.176·32-s − 1.20·34-s − 1/2·36-s + 0.657·37-s + 0.162·38-s + 0.158·40-s + 1.40·41-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9772220216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9772220216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.72596440893179193901180953608, −9.619536799040044852308641114462, −8.898354267006836038529810372650, −8.083404259994419740819564958376, −7.29647426292936217308034960880, −6.20838475694551267683654983785, −5.30241292046518562757702502308, −3.78309606096637116254138371365, −2.76199421744662066280568319074, −0.991840522862618277445081431089,
0.991840522862618277445081431089, 2.76199421744662066280568319074, 3.78309606096637116254138371365, 5.30241292046518562757702502308, 6.20838475694551267683654983785, 7.29647426292936217308034960880, 8.083404259994419740819564958376, 8.898354267006836038529810372650, 9.619536799040044852308641114462, 10.72596440893179193901180953608
