| L(s) = 1 | − 1.41i·5-s + 4i·7-s + 5.65·11-s − 4·13-s + 4.24i·17-s + 5.65·23-s + 2.99·25-s + 1.41i·29-s + 4i·31-s + 5.65·35-s + 6·37-s − 9.89i·41-s + 8i·43-s + 5.65·47-s − 9·49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s + 1.51i·7-s + 1.70·11-s − 1.10·13-s + 1.02i·17-s + 1.17·23-s + 0.599·25-s + 0.262i·29-s + 0.718i·31-s + 0.956·35-s + 0.986·37-s − 1.54i·41-s + 1.21i·43-s + 0.825·47-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44379 + 0.458892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44379 + 0.458892i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.92295975230878776874046135427, −9.636746367279832582119890311446, −8.988662126596628266130241397373, −8.521256754362502619770605852771, −7.15114571207234710370819670074, −6.18668713965497884219095361266, −5.27264475672384624893226264143, −4.30200732071547033856468519348, −2.87620669905022926389970862274, −1.51386925148197022456047979947,
1.00139004576048210737172999504, 2.82630491320167199391182977328, 3.98491503100421461092184165467, 4.81626122090875509237041380955, 6.41109217734208565932017327079, 7.07012598023383490879371606100, 7.64006565909740857192430892332, 9.147631667986383707719935815804, 9.733462023742142104423597225146, 10.69619063980368636171375991177
