L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 3·8-s + 9-s + 2·10-s − 4·11-s + 12-s − 2·15-s − 16-s + 17-s + 18-s + 4·19-s − 2·20-s − 4·22-s + 3·24-s − 25-s − 27-s − 2·29-s − 2·30-s + 8·31-s + 5·32-s + 4·33-s + 34-s − 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.852·22-s + 0.612·24-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.696·33-s + 0.171·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.777683275\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.777683275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.899563502118069928513035817023, −6.77134008945023385790272556073, −6.20220300959225106049479196342, −5.50102637582031221465989677425, −5.11362507524448207422057877871, −4.54155287812823127894903000452, −3.47376609518140105818965181337, −2.83406140322277320268959284899, −1.81715530933831301505475267050, −0.59589945071198317379558967988,
0.59589945071198317379558967988, 1.81715530933831301505475267050, 2.83406140322277320268959284899, 3.47376609518140105818965181337, 4.54155287812823127894903000452, 5.11362507524448207422057877871, 5.50102637582031221465989677425, 6.20220300959225106049479196342, 6.77134008945023385790272556073, 7.899563502118069928513035817023