L(s) = 1 | + (−0.863 + 0.280i)2-s + (0.587 + 0.809i)3-s + (−0.142 + 0.103i)4-s + (0.987 − 0.156i)5-s + (−0.734 − 0.533i)6-s + (0.627 − 0.863i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.412i)10-s + (−0.0966 − 0.297i)11-s + (−0.166 − 0.0542i)12-s + (0.951 + 0.309i)13-s + (0.707 + 0.707i)15-s + (−0.245 + 0.754i)16-s − 0.907i·18-s + (−0.124 + 0.124i)20-s + ⋯ |
L(s) = 1 | + (−0.863 + 0.280i)2-s + (0.587 + 0.809i)3-s + (−0.142 + 0.103i)4-s + (0.987 − 0.156i)5-s + (−0.734 − 0.533i)6-s + (0.627 − 0.863i)8-s + (−0.309 + 0.951i)9-s + (−0.809 + 0.412i)10-s + (−0.0966 − 0.297i)11-s + (−0.166 − 0.0542i)12-s + (0.951 + 0.309i)13-s + (0.707 + 0.707i)15-s + (−0.245 + 0.754i)16-s − 0.907i·18-s + (−0.124 + 0.124i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8730594017\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8730594017\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-0.987 + 0.156i)T \) |
| 13 | \( 1 + (-0.951 - 0.309i)T \) |
good | 2 | \( 1 + (0.863 - 0.280i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−10.13703309441723086968754845585, −9.458937325539296115617372699363, −8.688292081526480033072849009524, −8.382498619726031650440277038907, −7.21755748737326645978788851991, −6.18134253700859526935310597508, −5.13632042172916917043007711616, −4.14028364794565022191879672368, −3.11262136157304470970733054572, −1.64223146874534932071114887388,
1.27454563877749017762600835090, 2.13618708428655400869897223792, 3.31401957204235026711755517623, 4.89000495199002473330959848076, 5.96035712877150655204162892034, 6.71121767261732222303970976199, 7.86153182164594138201067113954, 8.406861197396729364424894436506, 9.348410496109021990120676971563, 9.701608108072480272508472201378