L(s) = 1 | + (−0.587 + 0.809i)5-s + (0.951 + 0.309i)9-s + (−0.142 − 0.278i)13-s + (0.278 + 1.76i)17-s + (−0.309 − 0.951i)25-s + (1.11 + 1.53i)29-s + (−0.809 + 0.412i)37-s + (0.363 − 1.11i)41-s + (−0.809 + 0.587i)45-s − i·49-s + (0.309 − 1.95i)53-s + (−0.363 − 1.11i)61-s + (0.309 + 0.0489i)65-s + (−1.76 − 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)5-s + (0.951 + 0.309i)9-s + (−0.142 − 0.278i)13-s + (0.278 + 1.76i)17-s + (−0.309 − 0.951i)25-s + (1.11 + 1.53i)29-s + (−0.809 + 0.412i)37-s + (0.363 − 1.11i)41-s + (−0.809 + 0.587i)45-s − i·49-s + (0.309 − 1.95i)53-s + (−0.363 − 1.11i)61-s + (0.309 + 0.0489i)65-s + (−1.76 − 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9267178027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9267178027\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
good | 3 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \) |
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
−10.31694008258101850962729621385, −10.23715734703461758251301394617, −8.719385907774051410384727739488, −7.997994621161512961444637380641, −7.10412936682212392158436898580, −6.45318271321102606408934226785, −5.20821098936728732443853037179, −4.08125792328088572479810977093, −3.26122588932964340226303580247, −1.78929370651795133692117373796,
1.10826721303962239185674361020, 2.81853348741585966095920596032, 4.21725264683786488437573627383, 4.72721646824320906932296344970, 5.92655307056831456375780830343, 7.14158405679597596495000805105, 7.66460833251440174787066464934, 8.758887418795856812831088854382, 9.487764890222658963501469369448, 10.17650746397232754371039149302