L(s) = 1 | + (−0.304 − 0.474i)5-s + (0.281 + 0.959i)9-s + (0.574 − 0.767i)13-s + (0.627 + 0.544i)17-s + (0.283 − 0.620i)25-s + (1.83 + 0.398i)29-s + (−1.41 − 1.41i)37-s + (0.959 + 1.28i)41-s + (0.368 − 0.425i)45-s + (−0.909 − 0.415i)49-s + (1.29 − 0.186i)53-s + (−0.398 − 0.148i)61-s + (−0.538 − 0.0385i)65-s + (0.273 + 0.0801i)73-s + (−0.841 + 0.540i)81-s + ⋯ |
L(s) = 1 | + (−0.304 − 0.474i)5-s + (0.281 + 0.959i)9-s + (0.574 − 0.767i)13-s + (0.627 + 0.544i)17-s + (0.283 − 0.620i)25-s + (1.83 + 0.398i)29-s + (−1.41 − 1.41i)37-s + (0.959 + 1.28i)41-s + (0.368 − 0.425i)45-s + (−0.909 − 0.415i)49-s + (1.29 − 0.186i)53-s + (−0.398 − 0.148i)61-s + (−0.538 − 0.0385i)65-s + (0.273 + 0.0801i)73-s + (−0.841 + 0.540i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.109085700\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109085700\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 + (0.841 + 0.540i)T \) |
good | 3 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 5 | \( 1 + (0.304 + 0.474i)T + (-0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (0.909 + 0.415i)T^{2} \) |
| 11 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.574 + 0.767i)T + (-0.281 - 0.959i)T^{2} \) |
| 17 | \( 1 + (-0.627 - 0.544i)T + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 23 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 0.398i)T + (0.909 + 0.415i)T^{2} \) |
| 31 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 37 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 41 | \( 1 + (-0.959 - 1.28i)T + (-0.281 + 0.959i)T^{2} \) |
| 43 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 47 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 53 | \( 1 + (-1.29 + 0.186i)T + (0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 61 | \( 1 + (0.398 + 0.148i)T + (0.755 + 0.654i)T^{2} \) |
| 67 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 79 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 97 | \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \) |
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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
−9.864842722631609553095531069910, −8.637304327174503604530468264479, −8.242396666668869828646447181878, −7.46026499519195032209488610995, −6.43755670070663672985683005590, −5.46452175794068922153004859450, −4.73146462612577929723291386068, −3.76404051691202706251974806923, −2.62342257719319214647635010841, −1.23310033476632181331293654967,
1.26821178756102249703269645693, 2.84923202894841296994521971791, 3.67599151260641255117820138670, 4.58387912385787175925784208679, 5.72443884798063038877140853571, 6.69816850960380746116020946638, 7.09463471076677640643764825485, 8.219744521017655625909167358417, 8.968653970849591312574582136299, 9.749167156205042607447885274850