L(s) = 1 | + (−2.08 + 3.60i)5-s + (18.2 − 2.97i)7-s + (−33.1 + 19.1i)11-s − 49.6i·13-s + (7.65 + 13.2i)17-s + (−122. − 70.9i)19-s + (136. + 78.9i)23-s + (53.8 + 93.2i)25-s + 204. i·29-s + (90.5 − 52.2i)31-s + (−27.3 + 72.1i)35-s + (−194. + 336. i)37-s − 325.·41-s − 191.·43-s + (249. − 432. i)47-s + ⋯ |
L(s) = 1 | + (−0.186 + 0.322i)5-s + (0.986 − 0.160i)7-s + (−0.909 + 0.524i)11-s − 1.05i·13-s + (0.109 + 0.189i)17-s + (−1.48 − 0.856i)19-s + (1.24 + 0.715i)23-s + (0.430 + 0.745i)25-s + 1.31i·29-s + (0.524 − 0.302i)31-s + (−0.131 + 0.348i)35-s + (−0.862 + 1.49i)37-s − 1.23·41-s − 0.678·43-s + (0.775 − 1.34i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.011117892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011117892\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-18.2 + 2.97i)T \) |
good | 5 | \( 1 + (2.08 - 3.60i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (33.1 - 19.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 49.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-7.65 - 13.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (122. + 70.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-136. - 78.9i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 204. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-90.5 + 52.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (194. - 336. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-249. + 432. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (37.8 - 21.8i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (86.5 + 149. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-208. - 120. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (440. + 763. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.01e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (361. - 208. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (237. - 411. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (298. - 517. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.77e3iT - 9.12e5T^{2} \) |
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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
−10.16046832407110906429605887247, −8.813360443027889967801499031454, −8.261056960963711510074226455600, −7.34655088173227792151623865003, −6.71642186043107513365306655503, −5.18990216277976469497697689202, −4.95917627739117931168564756083, −3.53136796419573901432647291050, −2.53284169736768899842645197554, −1.28069831226899701971417002270,
0.25179396297026974895483157530, 1.68504876076647502987462779589, 2.69634698190700312418199569346, 4.17969319509129912928386439497, 4.78299487331070625229849817772, 5.79581036113470366136805086589, 6.74698522699245296605707695335, 7.80571381436408923620519535862, 8.496863261740025371385081449266, 9.008444096941772702943131041926