L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 2·11-s − 13-s + 14-s + 16-s − 17-s + 4·19-s − 2·22-s + 7·23-s − 26-s + 28-s − 29-s + 3·31-s + 32-s − 34-s + 6·37-s + 4·38-s + 3·41-s + 43-s − 2·44-s + 7·46-s − 12·47-s + 49-s − 52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.426·22-s + 1.45·23-s − 0.196·26-s + 0.188·28-s − 0.185·29-s + 0.538·31-s + 0.176·32-s − 0.171·34-s + 0.986·37-s + 0.648·38-s + 0.468·41-s + 0.152·43-s − 0.301·44-s + 1.03·46-s − 1.75·47-s + 1/7·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.124854673\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.124854673\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.559445614310455916494758321079, −7.80516974580249205703440909183, −7.14253006005762024799153823515, −6.39860842539494741491816824731, −5.35395270183027751713079831508, −4.99694463086816364799307883145, −4.04206921003813732100393271148, −3.05783796505154958047042891710, −2.30836202040639030126924935645, −0.994854636871361997332992172547,
0.994854636871361997332992172547, 2.30836202040639030126924935645, 3.05783796505154958047042891710, 4.04206921003813732100393271148, 4.99694463086816364799307883145, 5.35395270183027751713079831508, 6.39860842539494741491816824731, 7.14253006005762024799153823515, 7.80516974580249205703440909183, 8.559445614310455916494758321079