L(s) = 1 | + (−0.0747 + 0.997i)3-s + (−0.365 + 0.930i)7-s + (−0.988 − 0.149i)9-s + (0.0931 + 0.116i)13-s + (0.365 + 0.632i)19-s + (−0.900 − 0.433i)21-s + (0.365 + 0.930i)25-s + (0.222 − 0.974i)27-s + (−0.733 + 1.26i)31-s + (−1.40 + 1.29i)37-s + (−0.123 + 0.0841i)39-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (−0.658 + 0.317i)57-s + (0.326 − 0.302i)61-s + ⋯ |
L(s) = 1 | + (−0.0747 + 0.997i)3-s + (−0.365 + 0.930i)7-s + (−0.988 − 0.149i)9-s + (0.0931 + 0.116i)13-s + (0.365 + 0.632i)19-s + (−0.900 − 0.433i)21-s + (0.365 + 0.930i)25-s + (0.222 − 0.974i)27-s + (−0.733 + 1.26i)31-s + (−1.40 + 1.29i)37-s + (−0.123 + 0.0841i)39-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (−0.658 + 0.317i)57-s + (0.326 − 0.302i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8551661228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8551661228\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0747 - 0.997i)T \) |
| 7 | \( 1 + (0.365 - 0.930i)T \) |
good | 5 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 19 | \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (0.733 - 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−9.493090790697534082215187820111, −8.749880039471963261096054220503, −8.306752472761293131041532461043, −7.05054575873653657031738137107, −6.26468386380332871275093496671, −5.30198579000323545210160062595, −4.98334127876693639700209609174, −3.57944145828076086601548165902, −3.18796396727950959929257972662, −1.84097077124193558520606011269,
0.56987836057923689590271086419, 1.86285574829364043728374985087, 2.96370648453313589758515948768, 3.88951260804115629707251228425, 4.98705602230301195426840108180, 5.89437948154302439856930705832, 6.72870569723742486906434895542, 7.21839624300453486578930546660, 7.970657010786165553963179515404, 8.719373301755938366123871317273