| L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s + 11-s + 6·13-s + 14-s − 16-s − 2·17-s + 4·19-s − 2·20-s + 22-s − 25-s + 6·26-s − 28-s + 2·29-s + 8·31-s + 5·32-s − 2·34-s + 2·35-s + 6·37-s + 4·38-s − 6·40-s − 10·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s − 0.948·40-s − 1.56·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.282665969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.282665969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−10.43992262323557902979025907081, −9.532486316872857827405438492086, −8.828553587531720858173448507294, −8.022234507412907531082436990413, −6.48470121230921862341444525516, −5.94382173518356535675711078051, −4.99234291175372167776627453992, −4.04378612288062447958179145910, −2.95767079823181201960858613834, −1.34955496402027160249830766749,
1.34955496402027160249830766749, 2.95767079823181201960858613834, 4.04378612288062447958179145910, 4.99234291175372167776627453992, 5.94382173518356535675711078051, 6.48470121230921862341444525516, 8.022234507412907531082436990413, 8.828553587531720858173448507294, 9.532486316872857827405438492086, 10.43992262323557902979025907081
