| L(s) = 1 | + (1.06 − 1.36i)3-s − 5-s − 1.51i·7-s + (−0.723 − 2.91i)9-s + 2.91·11-s − 1.06i·13-s + (−1.06 + 1.36i)15-s + 4.54i·17-s − 8.24·19-s + (−2.06 − 1.61i)21-s − 1.28i·23-s + 25-s + (−4.74 − 2.11i)27-s − 7.49i·29-s + 5.99i·31-s + ⋯ |
| L(s) = 1 | + (0.615 − 0.787i)3-s − 0.447·5-s − 0.571i·7-s + (−0.241 − 0.970i)9-s + 0.878·11-s − 0.296i·13-s + (−0.275 + 0.352i)15-s + 1.10i·17-s − 1.89·19-s + (−0.450 − 0.351i)21-s − 0.266i·23-s + 0.200·25-s + (−0.913 − 0.407i)27-s − 1.39i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7648468323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7648468323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.06 + 1.36i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (2.57 + 7.76i)T \) |
| good | 7 | \( 1 + 1.51iT - 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 1.06iT - 13T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 + 1.28iT - 23T^{2} \) |
| 29 | \( 1 + 7.49iT - 29T^{2} \) |
| 31 | \( 1 - 5.99iT - 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 + 3.43iT - 43T^{2} \) |
| 47 | \( 1 - 2.02iT - 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 + 8.92iT - 59T^{2} \) |
| 61 | \( 1 + 7.65iT - 61T^{2} \) |
| 71 | \( 1 + 1.51iT - 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 - 16.8iT - 79T^{2} \) |
| 83 | \( 1 + 9.82iT - 83T^{2} \) |
| 89 | \( 1 + 4.29iT - 89T^{2} \) |
| 97 | \( 1 - 16.1iT - 97T^{2} \) |
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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
−8.224005240794395371286384843955, −7.32770267132298763023505424671, −6.47609619745338901426345751872, −6.32403180839913947842597553007, −4.90885609532560974172739116352, −3.88240102521510788909985858747, −3.57239243255481521630354436438, −2.26312670134899174474930114483, −1.46624759097489731569268261092, −0.18885162993722120587604890275,
1.71146797006384747573039044740, 2.65383390059166963528129328579, 3.52648203951630179417667036936, 4.26400526511296878076380058880, 4.91084950655127902843401335270, 5.80370862390407573089418911817, 6.77294837437339156358421603432, 7.39721148012535268908957337955, 8.507616061746744302168694376495, 8.737683026150262717406704964310
