| L(s) = 1 | − 1.31i·5-s − 4.77·7-s + 3·9-s − 2.27i·11-s + 6.09i·13-s − 4.27·17-s + i·19-s − 3.46·23-s + 3.27·25-s + 6.09i·29-s + 2.62·31-s + 6.27i·35-s + 0.837i·37-s − 10.5·41-s + 10.2i·43-s + ⋯ |
| L(s) = 1 | − 0.587i·5-s − 1.80·7-s + 9-s − 0.685i·11-s + 1.68i·13-s − 1.03·17-s + 0.229i·19-s − 0.722·23-s + 0.654·25-s + 1.13i·29-s + 0.471·31-s + 1.06i·35-s + 0.137i·37-s − 1.64·41-s + 1.56i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7085206655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7085206655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 1.31iT - 5T^{2} \) |
| 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 + 2.27iT - 11T^{2} \) |
| 13 | \( 1 - 6.09iT - 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.09iT - 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 0.837iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 8.54iT - 59T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 - 8.54iT - 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 + 4.30T + 79T^{2} \) |
| 83 | \( 1 + 13.0iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−9.887924037948746166785394981526, −9.075411287014856530232978993717, −8.719253731781574709841305119728, −7.24416439246320593922193610161, −6.64829817154266407175693130257, −6.02632925095336370116787610235, −4.63224344237081132587438553932, −3.96680065718833501099000171312, −2.86706302912044422922297189904, −1.41934589845176613582732728408,
0.30353064188474975068237736203, 2.29672218191831991215928374594, 3.24542229861721662217447194214, 4.09006766000337156255922678264, 5.31922458184758662245751619950, 6.45176034654991702969123973796, 6.82882357071684882543076346199, 7.68851552477435339532462936424, 8.772385888071292859194367618119, 9.873934944705942616920154244955
