| L(s) = 1 | + (0.850 + 0.525i)2-s + (0.448 + 0.893i)4-s + (−0.745 + 0.666i)5-s + (−0.0878 + 0.996i)8-s + (0.492 − 0.870i)9-s + (−0.984 + 0.175i)10-s + (0.185 + 1.63i)13-s + (−0.597 + 0.801i)16-s + (0.0367 + 0.222i)17-s + (0.876 − 0.481i)18-s + (−0.929 − 0.368i)20-s + (0.112 − 0.993i)25-s + (−0.701 + 1.49i)26-s + (−0.0435 + 1.15i)29-s + (−0.929 + 0.368i)32-s + ⋯ |
| L(s) = 1 | + (0.850 + 0.525i)2-s + (0.448 + 0.893i)4-s + (−0.745 + 0.666i)5-s + (−0.0878 + 0.996i)8-s + (0.492 − 0.870i)9-s + (−0.984 + 0.175i)10-s + (0.185 + 1.63i)13-s + (−0.597 + 0.801i)16-s + (0.0367 + 0.222i)17-s + (0.876 − 0.481i)18-s + (−0.929 − 0.368i)20-s + (0.112 − 0.993i)25-s + (−0.701 + 1.49i)26-s + (−0.0435 + 1.15i)29-s + (−0.929 + 0.368i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.708882130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708882130\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.850 - 0.525i)T \) |
| 5 | \( 1 + (0.745 - 0.666i)T \) |
| good | 3 | \( 1 + (-0.492 + 0.870i)T^{2} \) |
| 7 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 11 | \( 1 + (-0.162 + 0.986i)T^{2} \) |
| 13 | \( 1 + (-0.185 - 1.63i)T + (-0.974 + 0.224i)T^{2} \) |
| 17 | \( 1 + (-0.0367 - 0.222i)T + (-0.947 + 0.320i)T^{2} \) |
| 19 | \( 1 + (0.910 + 0.414i)T^{2} \) |
| 23 | \( 1 + (-0.617 + 0.786i)T^{2} \) |
| 29 | \( 1 + (0.0435 - 1.15i)T + (-0.997 - 0.0753i)T^{2} \) |
| 31 | \( 1 + (0.947 - 0.320i)T^{2} \) |
| 37 | \( 1 + (1.88 - 0.584i)T + (0.823 - 0.567i)T^{2} \) |
| 41 | \( 1 + (-1.37 + 1.42i)T + (-0.0376 - 0.999i)T^{2} \) |
| 43 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 47 | \( 1 + (-0.899 - 0.437i)T^{2} \) |
| 53 | \( 1 + (-0.317 + 1.06i)T + (-0.837 - 0.546i)T^{2} \) |
| 59 | \( 1 + (0.863 + 0.503i)T^{2} \) |
| 61 | \( 1 + (-1.38 - 1.43i)T + (-0.0376 + 0.999i)T^{2} \) |
| 67 | \( 1 + (0.236 - 0.971i)T^{2} \) |
| 71 | \( 1 + (-0.693 + 0.720i)T^{2} \) |
| 73 | \( 1 + (-0.136 - 0.110i)T + (0.212 + 0.977i)T^{2} \) |
| 79 | \( 1 + (-0.492 + 0.870i)T^{2} \) |
| 83 | \( 1 + (-0.112 + 0.993i)T^{2} \) |
| 89 | \( 1 + (-0.396 + 0.258i)T + (0.402 - 0.915i)T^{2} \) |
| 97 | \( 1 + (-0.932 - 1.39i)T + (-0.379 + 0.925i)T^{2} \) |
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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
−9.052017429357551597922448759785, −8.552577573628040557399778780561, −7.45627780688089028299069190371, −6.88210723387293810181807347083, −6.53232888252475386310371915140, −5.47104215335365864494301283174, −4.37015723338135289202690666339, −3.87990388076530633832341866579, −3.11954756691512339631784124511, −1.85322866945317241505085323922,
0.886947313127545244054718283510, 2.17345356189266286428872025115, 3.25455238234379251334081414918, 4.03216596317581478063999020131, 4.93821856772910062370284940132, 5.37239588927639080095638057550, 6.35531609119124664760307755579, 7.50573992807225234357872090460, 7.893738375340914967538688978328, 8.866074792185677703401575920181
