L(s) = 1 | + (−0.796 − 1.16i)2-s + 3-s + (−0.731 + 1.86i)4-s + (−0.796 − 1.16i)6-s + 4.72i·7-s + (2.75 − 0.627i)8-s + 9-s − 3.93i·11-s + (−0.731 + 1.86i)12-s − 3.46·13-s + (5.51 − 3.76i)14-s + (−2.93 − 2.72i)16-s + 3.51i·17-s + (−0.796 − 1.16i)18-s + 5.44i·19-s + ⋯ |
L(s) = 1 | + (−0.563 − 0.826i)2-s + 0.577·3-s + (−0.365 + 0.930i)4-s + (−0.325 − 0.477i)6-s + 1.78i·7-s + (0.975 − 0.221i)8-s + 0.333·9-s − 1.18i·11-s + (−0.211 + 0.537i)12-s − 0.961·13-s + (1.47 − 1.00i)14-s + (−0.732 − 0.680i)16-s + 0.852i·17-s + (−0.187 − 0.275i)18-s + 1.24i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05553 + 0.377825i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05553 + 0.377825i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.796 + 1.16i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.72iT - 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.51iT - 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 - 0.414T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 + 0.925iT - 47T^{2} \) |
| 53 | \( 1 + 0.233T + 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.563iT - 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 - 7.27iT - 97T^{2} \) |
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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
−10.71401265428905046469263906538, −9.698272498844168564381436785469, −9.098821884025004954476814680971, −8.321833294709126184270493397968, −7.74323820052634336905524694511, −6.17217326917818870681427982343, −5.17929604465757827894595118964, −3.65401110249242816254299342341, −2.79445272217427513918392674720, −1.74808743734240588696126617749,
0.71989644200762941532033453472, 2.47916611851055270062720256990, 4.44599329893579208794227444084, 4.65802656610525912037347890212, 6.49954007998201195852845406426, 7.35152329066104432635264410570, 7.51970081355223431845068379030, 8.796978786736197962753822856328, 9.745993031381776759887104758159, 10.17513441834611318770602620148