L(s) = 1 | + 1.61·2-s − 1.90i·3-s + 1.61·4-s − 3.07i·6-s + 1.17i·7-s + 8-s − 2.61·9-s − 3.07i·12-s + 1.90i·14-s + (0.309 + 0.951i)17-s − 4.23·18-s + 2.23·21-s − 1.90i·24-s − 25-s + 3.07i·27-s + 1.90i·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s − 1.90i·3-s + 1.61·4-s − 3.07i·6-s + 1.17i·7-s + 8-s − 2.61·9-s − 3.07i·12-s + 1.90i·14-s + (0.309 + 0.951i)17-s − 4.23·18-s + 2.23·21-s − 1.90i·24-s − 25-s + 3.07i·27-s + 1.90i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.103670194\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.103670194\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.61T + T^{2} \) |
| 3 | \( 1 + 1.90iT - T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.17iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.90iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - 1.61T + T^{2} \) |
| 61 | \( 1 + 1.17iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.90iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.90iT - T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 - 1.17iT - T^{2} \) |
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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
−10.92730154303691863188511160998, −9.191939069420361291852238577934, −8.273583310429530167253117940190, −7.44494324473914114134352655656, −6.49927892204442259880984748444, −5.86076824090073874896331840795, −5.36466029016716930557473677270, −3.74545548360107902941522006729, −2.57705585961564947296051439386, −1.89740679322066617615386798472,
2.81711902218600475804408072674, 3.64964892789936085296986540618, 4.30614247235089430614961612031, 4.97094963084410077150673851390, 5.75878655530612182249045911703, 6.82735316295868090658360247170, 8.041441912268850933251627090273, 9.306409840874296298853476323030, 10.00773479758399957644495840651, 10.75542627810894982901629047608