L(s) = 1 | + (1.41 − 1.41i)5-s − 3i·9-s + (−4.24 − 4.24i)13-s − 2·17-s + 0.999i·25-s + (−7.07 − 7.07i)29-s + (1.41 − 1.41i)37-s + 10i·41-s + (−4.24 − 4.24i)45-s + 7·49-s + (9.89 − 9.89i)53-s + (−7.07 − 7.07i)61-s − 12·65-s − 6i·73-s − 9·81-s + ⋯ |
L(s) = 1 | + (0.632 − 0.632i)5-s − i·9-s + (−1.17 − 1.17i)13-s − 0.485·17-s + 0.199i·25-s + (−1.31 − 1.31i)29-s + (0.232 − 0.232i)37-s + 1.56i·41-s + (−0.632 − 0.632i)45-s + 49-s + (1.35 − 1.35i)53-s + (−0.905 − 0.905i)61-s − 1.48·65-s − 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311028777\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311028777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 5 | \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (4.24 + 4.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (7.07 + 7.07i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-9.89 + 9.89i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (7.07 + 7.07i)T + 61iT^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.620283036764787944307423816710, −9.064602094731346530248000317992, −8.037174266388778325042618447529, −7.20843442642607362112930182312, −6.11847789407009324111669185490, −5.44051043365300152293167439512, −4.48202280828293009755043746083, −3.29094604743136645679953673768, −2.09309480223514768698021602910, −0.56167569235541526640622874703,
1.93156024107026740741843752164, 2.60007298966074515473555153780, 4.08713117179827191216354712982, 5.04205152256619503955879385253, 5.91652349504064375793631567283, 7.05859567686355949057528065175, 7.38519512176941067332095039794, 8.718483851383060738656172169502, 9.377663218754535898129479680363, 10.33337060286031108206974089336