| L(s) = 1 | − 1.86i·3-s − 1.48i·5-s + 2.55·7-s − 0.484·9-s + 1.59i·11-s + i·13-s − 2.77·15-s + 4.76·17-s − 3.51i·19-s − 4.76i·21-s + 3.73·23-s + 2.79·25-s − 4.69i·27-s + 2i·29-s + 0.685·31-s + ⋯ |
| L(s) = 1 | − 1.07i·3-s − 0.664i·5-s + 0.964·7-s − 0.161·9-s + 0.479i·11-s + 0.277i·13-s − 0.715·15-s + 1.15·17-s − 0.806i·19-s − 1.03i·21-s + 0.778·23-s + 0.559·25-s − 0.903i·27-s + 0.371i·29-s + 0.123·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.109876855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109876855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.86iT - 3T^{2} \) |
| 5 | \( 1 + 1.48iT - 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 1.59iT - 11T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 + 3.51iT - 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 0.685T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 + 4.69iT - 43T^{2} \) |
| 47 | \( 1 + 5.38T + 47T^{2} \) |
| 53 | \( 1 - 5.52iT - 53T^{2} \) |
| 59 | \( 1 + 10.9iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 2.96iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 9.46T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 + 4.06iT - 83T^{2} \) |
| 89 | \( 1 - 0.969T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.946436033881004843746374001472, −8.314207168827939927355778213784, −7.45129165552161431740639602900, −7.05765714577829839926618403570, −5.94664090425854791031630649473, −4.99670318693973525976417797600, −4.38384221863674028192681088788, −2.88249339932109190646056009083, −1.68653237838944244783703347060, −0.983765087756470031980646848329,
1.34147868104949462142268910606, 2.87912147039192400376989343315, 3.64211030024093452397676517727, 4.60119900219469503302451664651, 5.33324247883696388052487346533, 6.18336561209684263026230328774, 7.34073904029902780441761124044, 7.984838690239413391764291525552, 8.838997379425023881722561023639, 9.733088762417380577911701743556
