| L(s) = 1 | − 1.53·5-s + (−0.414 + 2.61i)7-s − 0.585i·11-s + 2.16i·13-s − 5.86·17-s + 5.22i·19-s + 2.24i·23-s − 2.65·25-s + 5.41i·29-s + 4.32i·31-s + (0.634 − 4i)35-s + 4·37-s − 8.92·41-s + 10.4·43-s + 7.39·47-s + ⋯ |
| L(s) = 1 | − 0.684·5-s + (−0.156 + 0.987i)7-s − 0.176i·11-s + 0.600i·13-s − 1.42·17-s + 1.19i·19-s + 0.467i·23-s − 0.531·25-s + 1.00i·29-s + 0.777i·31-s + (0.107 − 0.676i)35-s + 0.657·37-s − 1.39·41-s + 1.59·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.423343 + 0.680887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423343 + 0.680887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.414 - 2.61i)T \) |
| good | 5 | \( 1 + 1.53T + 5T^{2} \) |
| 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 - 2.16iT - 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 5.22iT - 19T^{2} \) |
| 23 | \( 1 - 2.24iT - 23T^{2} \) |
| 29 | \( 1 - 5.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 - 4.58iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 5.86T + 89T^{2} \) |
| 97 | \( 1 + 8.28iT - 97T^{2} \) |
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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
−11.35725536794285145505649517467, −10.39463633835794441769344931868, −9.212471870285936984646406119646, −8.651523109667710336242848914061, −7.65248893490764274052872024141, −6.60074623450971334355498627667, −5.65859660778487579109339486427, −4.47182703082830680868300058183, −3.38351458115404639554851389148, −1.97840670040223002690160414132,
0.46453115106801834909575275425, 2.54430873637532515185982005033, 3.97496696422677786503810589583, 4.61900427521873195681166868718, 6.11540314734000902807377071659, 7.11984252221902802892324166353, 7.79268072139910048164236579171, 8.821147205023745614700458302962, 9.800644153738595449359495386805, 10.79323515505813204838016814777
