L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.675 + 1.17i)5-s − 0.351·7-s + (−0.499 − 0.866i)9-s − 5.52·11-s + (2.58 + 4.47i)13-s + (0.675 + 1.17i)15-s + (2.43 + 3.61i)19-s + (−0.175 + 0.304i)21-s + (4.41 + 7.63i)23-s + (1.58 + 2.74i)25-s − 0.999·27-s + (−1.35 − 2.34i)29-s + 0.524·31-s + (−2.76 + 4.78i)33-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.302 + 0.523i)5-s − 0.133·7-s + (−0.166 − 0.288i)9-s − 1.66·11-s + (0.717 + 1.24i)13-s + (0.174 + 0.302i)15-s + (0.559 + 0.828i)19-s + (−0.0383 + 0.0665i)21-s + (0.919 + 1.59i)23-s + (0.317 + 0.549i)25-s − 0.192·27-s + (−0.251 − 0.434i)29-s + 0.0941·31-s + (−0.480 + 0.832i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.941483 + 0.738285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.941483 + 0.738285i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-2.43 - 3.61i)T \) |
good | 5 | \( 1 + (0.675 - 1.17i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 0.351T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + (-2.58 - 4.47i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 7.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.35 + 2.34i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.524T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-1.35 + 2.34i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.26 - 5.65i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.02 - 3.51i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.76 - 4.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.938 - 1.62i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.99 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.52 + 4.37i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.85 - 6.67i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.91 + 6.77i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + (-2.32 - 4.02i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.90 + 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.26013723112315403751006893547, −9.419974006724268869879423167624, −8.485459141860654492998577620945, −7.60323463006521486676244594224, −7.10077601123258952476627604911, −6.01325263220962409329849966776, −5.10059652804735857263724118839, −3.71050838011052802892397165134, −2.88744657702586208518653963793, −1.58567693003855994962986159818,
0.55164722450051100403758847716, 2.60874542782096559274396435377, 3.37616086036901105644933970646, 4.82035851860667695339351919157, 5.18091671986280360337340663577, 6.43680761381306210675429554561, 7.66333827151791883220836374355, 8.300475888816108055494294514740, 8.931683677950106157258207620673, 10.05377638928067847505445476805