| L(s) = 1 | + (−1 + 1.41i)3-s − 0.585·5-s + 0.585·7-s + (−1.00 − 2.82i)9-s + 4.82i·11-s + (3 − 2i)13-s + (0.585 − 0.828i)15-s + 2.82i·17-s + 6.24·19-s + (−0.585 + 0.828i)21-s − 7.65·23-s − 4.65·25-s + (5.00 + 1.41i)27-s − 1.17i·29-s + 5.07·31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.816i)3-s − 0.261·5-s + 0.221·7-s + (−0.333 − 0.942i)9-s + 1.45i·11-s + (0.832 − 0.554i)13-s + (0.151 − 0.213i)15-s + 0.685i·17-s + 1.43·19-s + (−0.127 + 0.180i)21-s − 1.59·23-s − 0.931·25-s + (0.962 + 0.272i)27-s − 0.217i·29-s + 0.910·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9819215088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9819215088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 + 1.17iT - 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 - 9.65iT - 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8.82iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 8.82iT - 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 3.17iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4.48iT - 79T^{2} \) |
| 83 | \( 1 - 3.17iT - 83T^{2} \) |
| 89 | \( 1 + 8.58T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.917348864009673517107466130039, −9.572168491396811964057457415015, −8.265927919743381979306469702246, −7.72109672577492967364121053970, −6.48072801421408251101496233481, −5.80449359405535823943114197302, −4.77448940029541625188788860071, −4.11969293110976273186811240835, −3.10980225774699198379973735759, −1.45649566139343439212711711197,
0.47482921561119264598390732313, 1.73868977362867363481279899347, 3.11519061388371064251511633331, 4.18205961753310973219367231656, 5.53007184711374985178316116500, 5.91960106541200454248646014154, 6.93203680586600986080768054856, 7.77907977699537524706214087263, 8.380636170922796742749999067846, 9.292093644792309616462431646730
