| L(s) = 1 | − 2-s − 3·3-s + 4-s + 5-s + 3·6-s − 8-s + 6·9-s − 10-s + 4·11-s − 3·12-s + 6·13-s − 3·15-s + 16-s − 3·17-s − 6·18-s − 4·19-s + 20-s − 4·22-s + 2·23-s + 3·24-s − 4·25-s − 6·26-s − 9·27-s + 29-s + 3·30-s − 9·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.316·10-s + 1.20·11-s − 0.866·12-s + 1.66·13-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 1.41·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 0.417·23-s + 0.612·24-s − 4/5·25-s − 1.17·26-s − 1.73·27-s + 0.185·29-s + 0.547·30-s − 1.61·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.166370616944081000443573116224, −6.91709086173076802278430790865, −6.65289279907363962591023437374, −5.97094513512596786621599171042, −5.43732879203044208131785700285, −4.31263873378675555139071531297, −3.62649436742957430144265211200, −1.86916764810516353644919336689, −1.24683340114965282450007515780, 0,
1.24683340114965282450007515780, 1.86916764810516353644919336689, 3.62649436742957430144265211200, 4.31263873378675555139071531297, 5.43732879203044208131785700285, 5.97094513512596786621599171042, 6.65289279907363962591023437374, 6.91709086173076802278430790865, 8.166370616944081000443573116224
