L(s) = 1 | + (1 − 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (2.5 − 4.33i)5-s + 6·6-s + (0.809 + 18.5i)7-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (−5 − 8.66i)10-s + (21.9 + 37.9i)11-s + (6.00 − 10.3i)12-s + 66.8·13-s + (32.8 + 17.1i)14-s + 15.0·15-s + (−8 + 13.8i)16-s + (−8.07 − 13.9i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + (0.0437 + 0.999i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.601 + 1.04i)11-s + (0.144 − 0.249i)12-s + 1.42·13-s + (0.627 + 0.326i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.115 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.46776 + 0.132271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46776 + 0.132271i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
| 7 | \( 1 + (-0.809 - 18.5i)T \) |
good | 11 | \( 1 + (-21.9 - 37.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + (8.07 + 13.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-38.7 + 67.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (51.1 - 88.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-151. - 263. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-183. + 318. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 7.85T + 6.89e4T^{2} \) |
| 43 | \( 1 + 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (134. - 232. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (194. + 336. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (325. + 564. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (62.4 - 108. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-372. - 645. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 561.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-203. - 351. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-463. + 802. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (354. - 614. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 48.7T + 9.12e5T^{2} \) |
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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
−11.92383988813770411620319248971, −11.11058759692492252088446510359, −9.868811593413914835794482253898, −9.169571783698582260179572024801, −8.332779748308530954963856929002, −6.54904105253620163509834851718, −5.34658872254876761753605079223, −4.34212500033205738181498333907, −2.98626283765743198602633853329, −1.54974713627413221485302555176,
1.07968726238013717621673585583, 3.19641210301668849153500188750, 4.22515567297225110421096963391, 6.11424098561685173778580675136, 6.50338972284633379612834440254, 7.899960061065117581749789744864, 8.512717869820626408984389881979, 9.926592051407334494780842827901, 11.03179023696734559643749796190, 11.97310793418517651629693818700