L(s) = 1 | + (0.0921 + 1.41i)2-s + (−1.81 + 1.04i)3-s + (−1.98 + 0.260i)4-s + (0.894 + 0.894i)5-s + (−1.64 − 2.46i)6-s + (4.37 − 1.17i)7-s + (−0.549 − 2.77i)8-s + (0.693 − 1.20i)9-s + (−1.17 + 1.34i)10-s + (−0.404 + 1.51i)11-s + (3.32 − 2.54i)12-s + (−2.03 − 2.97i)13-s + (2.05 + 6.06i)14-s + (−2.55 − 0.685i)15-s + (3.86 − 1.03i)16-s + (−0.0484 − 0.0279i)17-s + ⋯ |
L(s) = 1 | + (0.0651 + 0.997i)2-s + (−1.04 + 0.604i)3-s + (−0.991 + 0.130i)4-s + (0.399 + 0.399i)5-s + (−0.671 − 1.00i)6-s + (1.65 − 0.442i)7-s + (−0.194 − 0.980i)8-s + (0.231 − 0.400i)9-s + (−0.372 + 0.425i)10-s + (−0.122 + 0.455i)11-s + (0.959 − 0.735i)12-s + (−0.565 − 0.824i)13-s + (0.549 + 1.61i)14-s + (−0.660 − 0.176i)15-s + (0.966 − 0.257i)16-s + (−0.0117 − 0.00678i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.424998 + 0.545154i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.424998 + 0.545154i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0921 - 1.41i)T \) |
| 13 | \( 1 + (2.03 + 2.97i)T \) |
good | 3 | \( 1 + (1.81 - 1.04i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.894 - 0.894i)T + 5iT^{2} \) |
| 7 | \( 1 + (-4.37 + 1.17i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.404 - 1.51i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.0484 + 0.0279i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.576 + 2.15i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.528 + 0.916i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.67 - 6.36i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.52 - 4.52i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.86 + 0.5i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.401 + 1.5i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.04 + 1.80i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.38 + 5.38i)T + 47iT^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 + (8.67 - 2.32i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.00 + 1.74i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.86 - 2.37i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.76 - 6.59i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (5.45 - 5.45i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.87iT - 79T^{2} \) |
| 83 | \( 1 + (-6.96 + 6.96i)T - 83iT^{2} \) |
| 89 | \( 1 + (15.3 + 4.10i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.440 - 0.118i)T + (84.0 - 48.5i)T^{2} \) |
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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
−15.85234760179041510366532707275, −14.76214536506127962630172511756, −14.02269120783984849641037885485, −12.40296510331885942119516875727, −10.93663963838438387540328554010, −10.12722190839460545423448356788, −8.330986230508667784957644520144, −7.02573671932970479975463391160, −5.40339185593007627962314478852, −4.65233643092870854138479557057,
1.71285655550302361469242236967, 4.74750394712209040073530888892, 5.81334217767118909951882546782, 7.992785898895246225670855774676, 9.358021224151850392706150250478, 11.02904579139956961548916061486, 11.64086547082700748343022207661, 12.49825298751140775535912827127, 13.77104903542041444907563120508, 14.79687556700536264610536400692