L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s + 4·11-s + 3·13-s − 14-s + 16-s + 17-s + 3·18-s − 4·22-s + 23-s − 3·26-s + 28-s − 4·31-s − 32-s − 34-s − 3·36-s − 11·37-s − 10·41-s + 2·43-s + 4·44-s − 46-s − 11·47-s + 49-s + 3·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 1.20·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 0.852·22-s + 0.208·23-s − 0.588·26-s + 0.188·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 1/2·36-s − 1.80·37-s − 1.56·41-s + 0.304·43-s + 0.603·44-s − 0.147·46-s − 1.60·47-s + 1/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.61266735752739277912609010135, −6.76017676956908874505703586832, −6.27755180380673637420577277825, −5.52315898744961489318093191762, −4.75539938563481692436701673467, −3.60086680211284785849130514803, −3.21149072381366395196915574082, −1.92640250190391300162941320261, −1.30191842673698294446155497649, 0,
1.30191842673698294446155497649, 1.92640250190391300162941320261, 3.21149072381366395196915574082, 3.60086680211284785849130514803, 4.75539938563481692436701673467, 5.52315898744961489318093191762, 6.27755180380673637420577277825, 6.76017676956908874505703586832, 7.61266735752739277912609010135