L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.951 − 1.30i)3-s + (0.809 − 0.587i)4-s + (1.30 + 0.951i)6-s + 0.618i·7-s + (−0.587 + 0.809i)8-s + (−0.500 + 1.53i)9-s + (−1.53 − 0.499i)12-s + (−0.190 − 0.587i)14-s + (0.309 − 0.951i)16-s − 1.61i·18-s + (0.809 − 0.587i)21-s + (0.587 − 0.190i)23-s + 1.61·24-s + (0.951 − 0.309i)27-s + (0.363 + 0.5i)28-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.951 − 1.30i)3-s + (0.809 − 0.587i)4-s + (1.30 + 0.951i)6-s + 0.618i·7-s + (−0.587 + 0.809i)8-s + (−0.500 + 1.53i)9-s + (−1.53 − 0.499i)12-s + (−0.190 − 0.587i)14-s + (0.309 − 0.951i)16-s − 1.61i·18-s + (0.809 − 0.587i)21-s + (0.587 − 0.190i)23-s + 1.61·24-s + (0.951 − 0.309i)27-s + (0.363 + 0.5i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4564509192\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4564509192\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 - 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61iT - T^{2} \) |
| 47 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 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
−9.055963261356702150620569376146, −8.213206931832018072593517696012, −7.55151710770728954778656829338, −6.92347637625078624421738945934, −6.20427256522231773134455046746, −5.67565892771135340591247028726, −4.84404891837948278550759068659, −2.99770992761247854593624583089, −1.96785565724303448412911428536, −1.07601507551550642549937349332,
0.55465601012455008581468849267, 2.18350695079351917998037033690, 3.65121571868478375625952231858, 3.98374595554413295570522174445, 5.21440899814518534742746220013, 5.86156811003681151998954893965, 6.91935961223733603207433643231, 7.48214315974859127189318848541, 8.653417034174895510203980200739, 9.179896099122135240160877751780