| L(s) = 1 | − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s − 4·11-s + 2·13-s − 4·14-s − 16-s − 2·17-s − 19-s + 20-s + 4·22-s + 4·23-s + 25-s − 2·26-s − 4·28-s + 2·29-s − 5·32-s + 2·34-s − 4·35-s − 6·37-s + 38-s − 3·40-s + 6·41-s + 8·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 0.371·29-s − 0.883·32-s + 0.342·34-s − 0.676·35-s − 0.986·37-s + 0.162·38-s − 0.474·40-s + 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9424032710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9424032710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 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 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.41920253312892486661363622049, −8.997665374443426177370400237956, −8.576861436588155713465515075006, −7.79111029211693930669614606595, −7.22090991652715915706448156971, −5.57442263951836682011583202205, −4.80915338219819536204131275847, −4.00775128266596845974693279239, −2.32830304228468473527730093464, −0.908717833826787650120905118138,
0.908717833826787650120905118138, 2.32830304228468473527730093464, 4.00775128266596845974693279239, 4.80915338219819536204131275847, 5.57442263951836682011583202205, 7.22090991652715915706448156971, 7.79111029211693930669614606595, 8.576861436588155713465515075006, 8.997665374443426177370400237956, 10.41920253312892486661363622049
