L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.697 − 0.0498i)5-s + (−0.841 + 0.540i)8-s + (0.989 − 0.142i)9-s + (0.494 + 0.494i)10-s + (−0.0683 − 0.125i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (0.755 + 0.654i)18-s + (−0.0498 + 0.697i)20-s + (−0.506 + 0.0727i)25-s + (0.0498 − 0.133i)26-s + (0.0801 + 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.697 − 0.0498i)5-s + (−0.841 + 0.540i)8-s + (0.989 − 0.142i)9-s + (0.494 + 0.494i)10-s + (−0.0683 − 0.125i)13-s + (−0.959 − 0.281i)16-s + (0.239 − 0.153i)17-s + (0.755 + 0.654i)18-s + (−0.0498 + 0.697i)20-s + (−0.506 + 0.0727i)25-s + (0.0498 − 0.133i)26-s + (0.0801 + 0.557i)29-s + (−0.415 − 0.909i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.716208148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.716208148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 353 | \( 1 + (-0.841 - 0.540i)T \) |
good | 3 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 5 | \( 1 + (-0.697 + 0.0498i)T + (0.989 - 0.142i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.0683 + 0.125i)T + (-0.540 + 0.841i)T^{2} \) |
| 17 | \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 23 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.0801 - 0.557i)T + (-0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 37 | \( 1 + (0.0903 - 0.415i)T + (-0.909 - 0.415i)T^{2} \) |
| 41 | \( 1 + (1.53 + 0.698i)T + (0.654 + 0.755i)T^{2} \) |
| 43 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 53 | \( 1 + (-0.0855 - 1.19i)T + (-0.989 + 0.142i)T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 73 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (0.148 + 0.682i)T + (-0.909 + 0.415i)T^{2} \) |
| 97 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
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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
−9.814199759689515828080208654489, −9.048515649487016814491676884345, −8.164133508596826849940656455589, −7.26666467067336139779210705796, −6.67011952648189075412368421853, −5.75828990523155719114623240239, −5.02552426303545266426350160696, −4.10936548859201459521017795683, −3.12489593977970240901648269693, −1.80129263157510118824719142101,
1.44154676304798174163277548108, 2.32698867437060913901012158724, 3.53348917786873847221825450377, 4.43848372479185458932825249525, 5.27120618788128573045444547080, 6.15418890429800418278868337073, 6.88957483011696029741459472693, 7.985438795035084867652861518259, 9.135669740325867967174003333701, 9.863380854322864510333547890105