L(s) = 1 | + (1.66 − 1.66i)3-s + (−0.707 − 0.707i)5-s − 2.89i·7-s − 2.53i·9-s + (−1.84 − 1.84i)11-s + (−3.08 + 3.08i)13-s − 2.35·15-s + 7.29·17-s + (1.23 − 1.23i)19-s + (−4.81 − 4.81i)21-s + 4.60i·23-s + 1.00i·25-s + (0.772 + 0.772i)27-s + (4.24 − 4.24i)29-s − 2.06·31-s + ⋯ |
L(s) = 1 | + (0.960 − 0.960i)3-s + (−0.316 − 0.316i)5-s − 1.09i·7-s − 0.845i·9-s + (−0.556 − 0.556i)11-s + (−0.854 + 0.854i)13-s − 0.607·15-s + 1.77·17-s + (0.283 − 0.283i)19-s + (−1.05 − 1.05i)21-s + 0.960i·23-s + 0.200i·25-s + (0.148 + 0.148i)27-s + (0.788 − 0.788i)29-s − 0.370·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15656 - 1.12575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15656 - 1.12575i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.66 + 1.66i)T - 3iT^{2} \) |
| 7 | \( 1 + 2.89iT - 7T^{2} \) |
| 11 | \( 1 + (1.84 + 1.84i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.08 - 3.08i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.60iT - 23T^{2} \) |
| 29 | \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + (1.17 + 1.17i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.61iT - 41T^{2} \) |
| 43 | \( 1 + (3.03 + 3.03i)T + 43iT^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.11 + 3.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \) |
| 67 | \( 1 + (8.24 - 8.24i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.25iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 0.113T + 79T^{2} \) |
| 83 | \( 1 + (9.76 - 9.76i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−11.63193870651332318699272988295, −10.36788122920074065530709023308, −9.445449533041121204342159960508, −8.289998499523385195261266504682, −7.57164565756589426183805735277, −7.04300248262253807273941995192, −5.43371941466018359175502185146, −3.99052167796275870186013046997, −2.79481921423873630186076275437, −1.17305401962085264527810958439,
2.58718927386562999807353433701, 3.32763729799747398611041380401, 4.76278795963669405666645809455, 5.69808063724828377799463987093, 7.38286540215188694257576241974, 8.208726837894893149372615434887, 9.083250906011074834326266638924, 10.06405993062870754838819534243, 10.49240469183657653447987829692, 12.19520201257881214668810607691