L(s) = 1 | + (1.41 − 1.73i)5-s − 4.89i·7-s − 3.46i·11-s + (−0.999 − 4.89i)25-s + 10.3i·29-s − 10·31-s + (−8.48 − 6.92i)35-s − 16.9·49-s + 14.1·53-s + (−5.99 − 4.89i)55-s + 10.3i·59-s − 9.79i·73-s − 16.9·77-s + 10·79-s + 5.65·83-s + ⋯ |
L(s) = 1 | + (0.632 − 0.774i)5-s − 1.85i·7-s − 1.04i·11-s + (−0.199 − 0.979i)25-s + 1.92i·29-s − 1.79·31-s + (−1.43 − 1.17i)35-s − 2.42·49-s + 1.94·53-s + (−0.809 − 0.660i)55-s + 1.35i·59-s − 1.14i·73-s − 1.93·77-s + 1.12·79-s + 0.620·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.593442329\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.593442329\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
good | 7 | \( 1 + 4.89iT - 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19.5iT - 97T^{2} \) |
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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
−9.149391909632970242267429015395, −8.580115013620057442193123971387, −7.53787740900187822290364337611, −6.91722019354896981780305752541, −5.87174692736383901084906681931, −5.05922089051886962017136704070, −4.08812953444809035284952661395, −3.28943928624846235276789453245, −1.62128297620107574046629900470, −0.63047446408072178130645993478,
2.06472893292625588889635423436, 2.41776521036936597091717866246, 3.70013735610571574852967408864, 5.05650150553598078445832468228, 5.72972707891141567591415584032, 6.43588585262053776106561951387, 7.34266705582853022693734802159, 8.266992419445714041228980262831, 9.280337106619467449776902529735, 9.590378136167846634026406152627