| L(s) = 1 | + (0.347 + 1.96i)2-s + (1.33 − 7.54i)3-s + (−3.75 + 1.36i)4-s + (−4.89 − 4.11i)5-s + 15.3·6-s + (−6.34 − 5.32i)7-s + (−4 − 6.92i)8-s + (−29.7 − 10.8i)9-s + (6.39 − 11.0i)10-s + (−32.5 − 56.4i)11-s + (5.32 + 30.1i)12-s + (24.8 − 9.04i)13-s + (8.28 − 14.3i)14-s + (−37.5 + 31.4i)15-s + (12.2 − 10.2i)16-s + (96.1 + 34.9i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.255 − 1.45i)3-s + (−0.469 + 0.171i)4-s + (−0.438 − 0.367i)5-s + 1.04·6-s + (−0.342 − 0.287i)7-s + (−0.176 − 0.306i)8-s + (−1.10 − 0.401i)9-s + (0.202 − 0.350i)10-s + (−0.893 − 1.54i)11-s + (0.127 + 0.725i)12-s + (0.530 − 0.192i)13-s + (0.158 − 0.273i)14-s + (−0.645 + 0.542i)15-s + (0.191 − 0.160i)16-s + (1.37 + 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.856437 - 0.910068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856437 - 0.910068i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.96i)T \) |
| 37 | \( 1 + (-93.2 - 204. i)T \) |
| good | 3 | \( 1 + (-1.33 + 7.54i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (4.89 + 4.11i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (6.34 + 5.32i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (32.5 + 56.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-24.8 + 9.04i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-96.1 - 34.9i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (11.5 - 65.7i)T + (-6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-41.4 + 71.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-113. - 196. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 94.8T + 2.97e4T^{2} \) |
| 41 | \( 1 + (37.3 - 13.5i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + 48.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-95.8 + 165. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-242. + 203. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-319. + 268. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-44.1 + 16.0i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-221. - 186. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-99.1 + 562. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + 955.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-396. - 332. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-1.20e3 - 439. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (143. - 120. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (617. - 1.06e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.66786032303429947293992046763, −12.93190568353233562965329253767, −12.07697229980123537484480719962, −10.46488176638156931920504228570, −8.370212132985523061436346755663, −8.095865075808238540418767471513, −6.70842568332052930600529914283, −5.60430989122729671552610116614, −3.34401219145602179006418089134, −0.789750672530324802677436403101,
2.84723755236318619636942137327, 4.12240852293076924280149951610, 5.30843067546048245230860909750, 7.54241195547217314951117385971, 9.217087121671031540142044993585, 9.923818211726799216093617745622, 10.84246867492124730497320144696, 11.96975600009950118841591541784, 13.20551747568506033539819865095, 14.61012337564956209178786557971
