| 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
−14.61012337564956209178786557971, −13.20551747568506033539819865095, −11.96975600009950118841591541784, −10.84246867492124730497320144696, −9.923818211726799216093617745622, −9.217087121671031540142044993585, −7.54241195547217314951117385971, −5.30843067546048245230860909750, −4.12240852293076924280149951610, −2.84723755236318619636942137327,
0.789750672530324802677436403101, 3.34401219145602179006418089134, 5.60430989122729671552610116614, 6.70842568332052930600529914283, 8.095865075808238540418767471513, 8.370212132985523061436346755663, 10.46488176638156931920504228570, 12.07697229980123537484480719962, 12.93190568353233562965329253767, 13.66786032303429947293992046763
