L(s) = 1 | + (−1.49 + 0.215i)5-s + (0.755 + 0.654i)9-s + (0.0498 − 0.133i)13-s + (−1.03 + 1.61i)17-s + (1.23 − 0.361i)25-s + (−1.05 + 1.40i)29-s + (0.300 + 0.300i)37-s + (0.654 + 1.75i)41-s + (−1.27 − 0.817i)45-s + (0.281 + 0.959i)49-s + (1.53 − 0.698i)53-s + (−1.40 + 0.767i)61-s + (−0.0458 + 0.210i)65-s + (−0.544 − 0.627i)73-s + (0.142 + 0.989i)81-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.215i)5-s + (0.755 + 0.654i)9-s + (0.0498 − 0.133i)13-s + (−1.03 + 1.61i)17-s + (1.23 − 0.361i)25-s + (−1.05 + 1.40i)29-s + (0.300 + 0.300i)37-s + (0.654 + 1.75i)41-s + (−1.27 − 0.817i)45-s + (0.281 + 0.959i)49-s + (1.53 − 0.698i)53-s + (−1.40 + 0.767i)61-s + (−0.0458 + 0.210i)65-s + (−0.544 − 0.627i)73-s + (0.142 + 0.989i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6908950181\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6908950181\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
good | 3 | \( 1 + (-0.755 - 0.654i)T^{2} \) |
| 5 | \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \) |
| 7 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 11 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.0498 + 0.133i)T + (-0.755 - 0.654i)T^{2} \) |
| 17 | \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 23 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 29 | \( 1 + (1.05 - 1.40i)T + (-0.281 - 0.959i)T^{2} \) |
| 31 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 37 | \( 1 + (-0.300 - 0.300i)T + iT^{2} \) |
| 41 | \( 1 + (-0.654 - 1.75i)T + (-0.755 + 0.654i)T^{2} \) |
| 43 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 47 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 53 | \( 1 + (-1.53 + 0.698i)T + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 61 | \( 1 + (1.40 - 0.767i)T + (0.540 - 0.841i)T^{2} \) |
| 67 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 73 | \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 97 | \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
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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
−10.10860092667315537835808512612, −8.940109752504525278500719612297, −8.236014867859257152866761378467, −7.55377080545404138179569813135, −6.92394381727672017631817094388, −5.88363672831284221770243379129, −4.57594541851334834685735514848, −4.11342115810844456281937827692, −3.10538443225973502443100912899, −1.65185901491239214099501731411,
0.58643323335842644552160719415, 2.41585713002446961384654983641, 3.78815673265911605944893223409, 4.19989755863485579727296931843, 5.19251463615621447538831100381, 6.46237982523685797215416724450, 7.33266864265883322184000287051, 7.67786992394814475597154044135, 8.888146700247714618274694697059, 9.290647034487193865707894798203