L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 2·7-s − 3·9-s − 4·10-s − 3·11-s + 4·13-s − 4·14-s − 4·16-s + 17-s − 6·18-s − 19-s − 4·20-s − 6·22-s + 23-s − 25-s + 8·26-s − 4·28-s − 6·29-s − 8·31-s − 8·32-s + 2·34-s + 4·35-s − 6·36-s − 2·37-s − 2·38-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 0.755·7-s − 9-s − 1.26·10-s − 0.904·11-s + 1.10·13-s − 1.06·14-s − 16-s + 0.242·17-s − 1.41·18-s − 0.229·19-s − 0.894·20-s − 1.27·22-s + 0.208·23-s − 1/5·25-s + 1.56·26-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s + 0.676·35-s − 36-s − 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−9.491089311396895428526430046218, −8.643308621193790780325039006644, −7.76127111271616858613177130690, −6.76143361930033746876890443837, −5.78395595155708678947454558582, −5.32342147634490412558056408515, −3.93529221306480405675404880745, −3.51670357488875386563464505592, −2.50500987736200068031649270207, 0,
2.50500987736200068031649270207, 3.51670357488875386563464505592, 3.93529221306480405675404880745, 5.32342147634490412558056408515, 5.78395595155708678947454558582, 6.76143361930033746876890443837, 7.76127111271616858613177130690, 8.643308621193790780325039006644, 9.491089311396895428526430046218