| L(s) = 1 | + 2-s − 4-s + 2·5-s + 4·7-s − 3·8-s + 2·10-s − 2·13-s + 4·14-s − 16-s − 17-s − 4·19-s − 2·20-s − 4·23-s − 25-s − 2·26-s − 4·28-s − 6·29-s + 4·31-s + 5·32-s − 34-s + 8·35-s − 2·37-s − 4·38-s − 6·40-s + 6·41-s + 4·43-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.171·34-s + 1.35·35-s − 0.328·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s + 0.609·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.585253218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585253218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.13887678046194086016444609016, −12.16602218247269112718036034946, −11.11422065032987731085727997885, −9.912917364966795607879749961117, −8.872689562844898968412116321231, −7.81886564473676875900189501384, −6.12380991478521668699719766019, −5.15238580447119340700000765704, −4.18976898855068708054002343552, −2.14503914986953664301144200995,
2.14503914986953664301144200995, 4.18976898855068708054002343552, 5.15238580447119340700000765704, 6.12380991478521668699719766019, 7.81886564473676875900189501384, 8.872689562844898968412116321231, 9.912917364966795607879749961117, 11.11422065032987731085727997885, 12.16602218247269112718036034946, 13.13887678046194086016444609016
