| L(s) = 1 | − 2·4-s − 2·5-s − 3·7-s + 13-s + 4·16-s + 2·17-s − 7·19-s + 4·20-s − 8·23-s − 25-s + 6·28-s − 6·29-s + 5·31-s + 6·35-s − 3·37-s + 4·41-s + 4·43-s − 6·47-s + 2·49-s − 2·52-s + 4·53-s − 6·59-s − 11·61-s − 8·64-s − 2·65-s − 3·67-s − 4·68-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 1.13·7-s + 0.277·13-s + 16-s + 0.485·17-s − 1.60·19-s + 0.894·20-s − 1.66·23-s − 1/5·25-s + 1.13·28-s − 1.11·29-s + 0.898·31-s + 1.01·35-s − 0.493·37-s + 0.624·41-s + 0.609·43-s − 0.875·47-s + 2/7·49-s − 0.277·52-s + 0.549·53-s − 0.781·59-s − 1.40·61-s − 64-s − 0.248·65-s − 0.366·67-s − 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14157 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−16.54887360507324, −16.19741471483877, −15.61808223442100, −14.96232578529806, −14.57780352268707, −13.68791085057380, −13.43325929288581, −12.72333396840047, −12.23283106647308, −11.89767435264773, −10.82138112923136, −10.49712083564882, −9.568468241754712, −9.470482666927445, −8.484985666349242, −8.120428442113119, −7.558163541203709, −6.635377147259025, −6.051731380705423, −5.498484340593158, −4.364837967335280, −4.086773010138506, −3.505757316123722, −2.667884081482667, −1.479002360313681, 0, 0,
1.479002360313681, 2.667884081482667, 3.505757316123722, 4.086773010138506, 4.364837967335280, 5.498484340593158, 6.051731380705423, 6.635377147259025, 7.558163541203709, 8.120428442113119, 8.484985666349242, 9.470482666927445, 9.568468241754712, 10.49712083564882, 10.82138112923136, 11.89767435264773, 12.23283106647308, 12.72333396840047, 13.43325929288581, 13.68791085057380, 14.57780352268707, 14.96232578529806, 15.61808223442100, 16.19741471483877, 16.54887360507324
