L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61i·7-s + (0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.587 − 0.190i)12-s + (0.500 + 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618i·18-s + (0.809 − 0.587i)21-s + (1.53 − 0.5i)23-s − 0.618·24-s + (−0.951 + 0.309i)27-s + (0.951 + 1.30i)28-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (−0.5 − 0.363i)6-s + 1.61i·7-s + (0.587 − 0.809i)8-s + (0.190 − 0.587i)9-s + (−0.587 − 0.190i)12-s + (0.500 + 1.53i)14-s + (0.309 − 0.951i)16-s − 0.618i·18-s + (0.809 − 0.587i)21-s + (1.53 − 0.5i)23-s − 0.618·24-s + (−0.951 + 0.309i)27-s + (0.951 + 1.30i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.038281062\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.038281062\) |
\(L(1)\) |
|
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 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 - 1.61iT - T^{2} \) |
| 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 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 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 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.134243028353035364487516267976, −8.209882019883114131649714945682, −7.10717537580601376829572731898, −6.50197103788852191967605258161, −5.80803069247723096205514599945, −5.21494761690833408035255513524, −4.27734163056205190914135954593, −3.10586889761170867483202245624, −2.43411741747403841401146720769, −1.27152748627914436400486702395,
1.45099762011790078038706254524, 2.91801339883104498661046554793, 3.81471589214732982108705302363, 4.53745996726259665182123444342, 5.06198705789343238629797757092, 5.99901059458384041059656834966, 7.02931525235320627353719705252, 7.34337659954529163508196517041, 8.163910923326706358926327529069, 9.256131188551315450049071296363