L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s − 3·9-s + 10-s + 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s + 4·19-s + 20-s − 4·23-s + 25-s + 6·26-s − 28-s − 6·29-s + 32-s + 2·34-s − 35-s − 3·36-s − 2·37-s + 4·38-s + 40-s + 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.917·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 1/2·36-s − 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.595786744\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.595786744\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.78756367385159341430981998936, −6.92083137647374604242018824744, −6.12961997921981853635910763631, −5.74392610329482960040574163447, −5.26964949099340810286657963874, −4.06253115986344144512606759561, −3.54212688670020728693570411453, −2.82968211807009410797264506076, −1.90897112784761245945570013279, −0.844515960079188780436889620071,
0.844515960079188780436889620071, 1.90897112784761245945570013279, 2.82968211807009410797264506076, 3.54212688670020728693570411453, 4.06253115986344144512606759561, 5.26964949099340810286657963874, 5.74392610329482960040574163447, 6.12961997921981853635910763631, 6.92083137647374604242018824744, 7.78756367385159341430981998936