| L(s) = 1 | − 1.93·2-s + (1.41 − i)3-s + 1.73·4-s + 1.41i·5-s + (−2.73 + 1.93i)6-s − 3.73·7-s + 0.517·8-s + (1.00 − 2.82i)9-s − 2.73i·10-s + 0.378i·11-s + (2.44 − 1.73i)12-s + 3.73i·13-s + 7.20·14-s + (1.41 + 2.00i)15-s − 4.46·16-s − 4.89i·17-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + (0.816 − 0.577i)3-s + 0.866·4-s + 0.632i·5-s + (−1.11 + 0.788i)6-s − 1.41·7-s + 0.183·8-s + (0.333 − 0.942i)9-s − 0.863i·10-s + 0.114i·11-s + (0.707 − 0.500i)12-s + 1.03i·13-s + 1.92·14-s + (0.365 + 0.516i)15-s − 1.11·16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8357576774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8357576774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.41 + i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 3.73iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 23 | \( 1 - 0.378iT - 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 - 4.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.26iT - 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 3.53iT - 79T^{2} \) |
| 83 | \( 1 - 7.72iT - 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 7.46iT - 97T^{2} \) |
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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
−9.671399975609153168497858338172, −9.135951007999722987586010423977, −8.459908340041575549253553004112, −7.36149327233128748043741357134, −6.90229395590209562915309180418, −6.33835985873000227747001856178, −4.47321292118984896270712866558, −3.14469359348905339811997397112, −2.43627353843630021346637553457, −0.950338124943267235274153746016,
0.71592683505890323654087351326, 2.32209306025843171714849311561, 3.42212741904663672061153298199, 4.42766046250668612550668937065, 5.68415380520686101665804573408, 6.80080872045812888635082967435, 7.78478338059278520092331882839, 8.489028034072754389484252867630, 8.973962671208776569122799439692, 9.763745228620356902214127452225
