| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 5·11-s − 14-s + 16-s − 2·17-s − 19-s − 5·22-s + 23-s + 28-s + 4·29-s − 9·31-s − 32-s + 2·34-s − 5·37-s + 38-s − 9·41-s + 10·43-s + 5·44-s − 46-s − 6·47-s + 49-s − 12·53-s − 56-s − 4·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.50·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s − 1.06·22-s + 0.208·23-s + 0.188·28-s + 0.742·29-s − 1.61·31-s − 0.176·32-s + 0.342·34-s − 0.821·37-s + 0.162·38-s − 1.40·41-s + 1.52·43-s + 0.753·44-s − 0.147·46-s − 0.875·47-s + 1/7·49-s − 1.64·53-s − 0.133·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.35812576798958008272532883170, −6.72730401761252630813038890466, −6.26063306113512207795588921958, −5.35884604513418383479068967446, −4.52751670540137898503199424631, −3.78368358517150033170139625394, −2.97877578986197237673695419207, −1.84625883617289406032279622661, −1.33324742484933982886227459461, 0,
1.33324742484933982886227459461, 1.84625883617289406032279622661, 2.97877578986197237673695419207, 3.78368358517150033170139625394, 4.52751670540137898503199424631, 5.35884604513418383479068967446, 6.26063306113512207795588921958, 6.72730401761252630813038890466, 7.35812576798958008272532883170
