L(s) = 1 | + (0.433 + 1.34i)2-s + (0.497 − 0.497i)3-s + (−1.62 + 1.16i)4-s + (−2.01 + 0.978i)5-s + (0.885 + 0.454i)6-s + (−2.91 − 2.91i)7-s + (−2.27 − 1.68i)8-s + 2.50i·9-s + (−2.18 − 2.28i)10-s + 3.58i·11-s + (−0.227 + 1.38i)12-s + (−3.29 − 3.29i)13-s + (2.66 − 5.19i)14-s + (−0.514 + 1.48i)15-s + (1.27 − 3.79i)16-s + (−4.60 + 4.60i)17-s + ⋯ |
L(s) = 1 | + (0.306 + 0.951i)2-s + (0.287 − 0.287i)3-s + (−0.812 + 0.583i)4-s + (−0.899 + 0.437i)5-s + (0.361 + 0.185i)6-s + (−1.10 − 1.10i)7-s + (−0.804 − 0.594i)8-s + 0.834i·9-s + (−0.691 − 0.721i)10-s + 1.08i·11-s + (−0.0658 + 0.401i)12-s + (−0.914 − 0.914i)13-s + (0.712 − 1.38i)14-s + (−0.132 + 0.384i)15-s + (0.319 − 0.947i)16-s + (−1.11 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0982257 - 0.367782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0982257 - 0.367782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.433 - 1.34i)T \) |
| 5 | \( 1 + (2.01 - 0.978i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (-0.497 + 0.497i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.91 + 2.91i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + (3.29 + 3.29i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.60 - 4.60i)T - 17iT^{2} \) |
| 23 | \( 1 + (-3.51 + 3.51i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.88iT - 29T^{2} \) |
| 31 | \( 1 - 1.57iT - 31T^{2} \) |
| 37 | \( 1 + (5.48 - 5.48i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + (-4.56 + 4.56i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.60 - 1.60i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.12 - 2.12i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 - 0.536T + 61T^{2} \) |
| 67 | \( 1 + (-1.64 - 1.64i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.08iT - 71T^{2} \) |
| 73 | \( 1 + (0.480 + 0.480i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 + (12.4 - 12.4i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + (6.21 - 6.21i)T - 97iT^{2} \) |
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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
−12.43115890292766733646747476628, −10.68350111010720150812562794598, −10.18876341614787356509278844434, −8.798551984521010226568325528331, −7.84616585021370637850952364010, −7.08717285676843275189770675124, −6.67613263514807765907788699331, −4.94318815832860417289725973586, −4.05682173374237597568648836963, −2.89047460264024855989615345418,
0.21250072696902941684987762596, 2.61329434974595857038640734447, 3.49759308970697710549361512331, 4.54391820042659270778134647696, 5.74074711060240252081520162035, 6.89945833763180300365639088155, 8.612015523805383508557256222433, 9.172060303570476698673659286026, 9.638612703452057604104760311166, 11.17017080806762148616092574431