L(s) = 1 | + (−1.73 + i)2-s + (−0.261 + 0.452i)3-s + (1.99 − 3.46i)4-s + (2.77 + 1.60i)5-s − 1.04i·6-s + (8.06 − 13.9i)7-s + 7.99i·8-s + (13.3 + 23.1i)9-s − 6.40·10-s + 4.89·11-s + (1.04 + 1.81i)12-s + (71.2 + 41.1i)13-s + 32.2i·14-s + (−1.44 + 0.836i)15-s + (−8 − 13.8i)16-s + (−16.5 + 9.56i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.0502 + 0.0870i)3-s + (0.249 − 0.433i)4-s + (0.247 + 0.143i)5-s − 0.0711i·6-s + (0.435 − 0.753i)7-s + 0.353i·8-s + (0.494 + 0.857i)9-s − 0.202·10-s + 0.134·11-s + (0.0251 + 0.0435i)12-s + (1.51 + 0.877i)13-s + 0.615i·14-s + (−0.0249 + 0.0143i)15-s + (−0.125 − 0.216i)16-s + (−0.236 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.19124 + 0.399148i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19124 + 0.399148i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.73 - i)T \) |
| 37 | \( 1 + (221. + 38.3i)T \) |
good | 3 | \( 1 + (0.261 - 0.452i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (-2.77 - 1.60i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-8.06 + 13.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 - 4.89T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-71.2 - 41.1i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (16.5 - 9.56i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-99.0 - 57.1i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 27.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 100. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 229. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (209. - 362. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + 241. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-2.68 - 4.65i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (441. - 255. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-111. - 64.4i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-188. + 327. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (182. - 315. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 70.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + (294. + 170. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (561. + 973. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (867. - 500. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 483. iT - 9.12e5T^{2} \) |
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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
−13.98231517517571459088418844374, −13.56700349532113358976622729244, −11.64169143346528303159209813772, −10.70053380131299058269065849442, −9.728239307919077383314443630905, −8.321742640091239192051707066441, −7.28173982761336328270570326443, −5.92000329697329237071234108223, −4.18798512035705343873677262562, −1.56151947570836324572018052965,
1.28842771989462858954683984972, 3.37073896706378868671092251157, 5.48369789285967062321031732620, 6.96337151170224169660199304928, 8.507824268571637421713061226947, 9.283690185087571512256862059769, 10.62426961013409421394784450406, 11.70491245133004480275075440893, 12.66435835498133927154332819036, 13.80152651423057869322266858201