L(s) = 1 | − 2·2-s + (3.01 − 4.23i)3-s + 4·4-s − 19.8i·5-s + (−6.02 + 8.46i)6-s + 26.1·7-s − 8·8-s + (−8.82 − 25.5i)9-s + 39.6i·10-s − 40.5·11-s + (12.0 − 16.9i)12-s − 74.9i·13-s − 52.3·14-s + (−83.8 − 59.6i)15-s + 16·16-s + 1.85i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.580 − 0.814i)3-s + 0.5·4-s − 1.77i·5-s + (−0.410 + 0.575i)6-s + 1.41·7-s − 0.353·8-s + (−0.327 − 0.945i)9-s + 1.25i·10-s − 1.11·11-s + (0.290 − 0.407i)12-s − 1.59i·13-s − 0.998·14-s + (−1.44 − 1.02i)15-s + 0.250·16-s + 0.0265i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.731888735\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731888735\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (-3.01 + 4.23i)T \) |
| 59 | \( 1 + (381. + 244. i)T \) |
good | 5 | \( 1 + 19.8iT - 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 11 | \( 1 + 40.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 1.85iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 80.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 227. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 52.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 275. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 132. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 85.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 503.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 277. iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 770. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 84.3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 188. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 169. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 47.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 466.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 577. iT - 9.12e5T^{2} \) |
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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
−10.61220347119122257723466758704, −9.344657661698287215869528672572, −8.522607235257423950625673176139, −8.026041091841724265987585851332, −7.40873893196561298148495140378, −5.51024703463155426245499887506, −4.96057060973993865038300028304, −2.95583760820473103092866738156, −1.44718348242414579260039927939, −0.75733935374510750052662402660,
2.06673098254842790679904100705, 2.90664950409021571964173750108, 4.30260781154432584856259605114, 5.60216017282583690206777727438, 7.17591587619298188637870345125, 7.61064662735860092776736135577, 8.730780955610419596095062244213, 9.674399429412496794150062796296, 10.61733017591953126131482184041, 11.08699882702028390641624991069