L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 3·9-s − 4·11-s + 6·13-s + 14-s + 16-s + 3·18-s + 4·22-s − 6·26-s − 28-s − 6·29-s − 8·31-s − 32-s − 3·36-s − 10·37-s − 2·41-s − 4·43-s − 4·44-s − 8·47-s + 49-s + 6·52-s + 2·53-s + 56-s + 6·58-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 + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.852·22-s − 1.17·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1/2·36-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 0.274·53-s + 0.133·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.73861738998774, −13.66527486572513, −12.87768478162605, −12.67225987203376, −11.90846904150114, −11.28207429611528, −11.00940894107766, −10.68527481251396, −10.02967133409269, −9.491596104472005, −8.992908627962992, −8.396169656233916, −8.245424059381933, −7.614379083133301, −6.903174094456593, −6.518312879597412, −5.841817830324479, −5.370916474048445, −5.061080838504499, −3.700239814210280, −3.625709909790842, −2.948156658945282, −2.135816737401968, −1.693926953094215, −0.6537340265133293, 0,
0.6537340265133293, 1.693926953094215, 2.135816737401968, 2.948156658945282, 3.625709909790842, 3.700239814210280, 5.061080838504499, 5.370916474048445, 5.841817830324479, 6.518312879597412, 6.903174094456593, 7.614379083133301, 8.245424059381933, 8.396169656233916, 8.992908627962992, 9.491596104472005, 10.02967133409269, 10.68527481251396, 11.00940894107766, 11.28207429611528, 11.90846904150114, 12.67225987203376, 12.87768478162605, 13.66527486572513, 13.73861738998774