| L(s) = 1 | − i·2-s − 4-s + (2 + 1.73i)7-s + i·8-s + (2.59 − 1.5i)11-s + (−3 + 1.73i)13-s + (1.73 − 2i)14-s + 16-s + (−1.5 − 2.59i)22-s + (2.5 + 4.33i)25-s + (1.73 + 3i)26-s + (−2 − 1.73i)28-s + (7.79 + 4.5i)29-s + 1.73i·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.755 + 0.654i)7-s + 0.353i·8-s + (0.783 − 0.452i)11-s + (−0.832 + 0.480i)13-s + (0.462 − 0.534i)14-s + 0.250·16-s + (−0.319 − 0.553i)22-s + (0.5 + 0.866i)25-s + (0.339 + 0.588i)26-s + (−0.377 − 0.327i)28-s + (1.44 + 0.835i)29-s + 0.311i·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.675323556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675323556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.79 - 4.5i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.19 + 9i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + (-2.59 + 4.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.5 + 4.33i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.795547396830284871130690390313, −8.932111992821338437990276986026, −8.496358154560628141973629987167, −7.36398989571440720591351731224, −6.40183391170559491877733701393, −5.24978166791980764307570772754, −4.61307074983977117769119259934, −3.43220125149618344050360414441, −2.37036933212909631429203548849, −1.23826749611413355540410311863,
0.896873111094220685163242369539, 2.49494748670188363720864102726, 4.06054966699887364616293294041, 4.62507169351894128553977253596, 5.61979285213618131718016219272, 6.67592312339143422220367775459, 7.32004473487359547531605572613, 8.113801916601002756922406720199, 8.829781906470374448772978908756, 9.932395325517301567024650269107
