L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 − 1.73i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−1.5 − 2.59i)11-s − 5·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (1.5 + 2.59i)18-s + (2.5 − 4.33i)19-s + 3·22-s + (3.5 − 6.06i)23-s + (2.5 − 4.33i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.452 − 0.783i)11-s − 1.38·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (0.353 + 0.612i)18-s + (0.573 − 0.993i)19-s + 0.639·22-s + (0.729 − 1.26i)23-s + (0.490 − 0.849i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.633988 - 0.421717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.633988 - 0.421717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-8 - 13.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−11.11584165163924703177341758099, −10.13474066569265854156429223107, −9.510857966204882455824488225416, −8.532657919845425121242453096632, −7.25978192377663568102751808188, −6.79614066558429530228927765220, −5.58120439354575535628816104719, −4.34548316261812174988840844362, −2.96673359541490778232284975198, −0.58332894184136697956191633983,
1.98642303142159471363102641943, 3.10073157014197502721780883375, 4.65124743374832625105413764831, 5.62233238596070663715367089269, 7.32382579857964788089426536430, 7.72716939365291428698653931590, 9.327750071305260772839371146644, 9.724475603072259579790942875550, 10.58729930652192783056899494311, 11.74212238638542392568587489661