| L(s) = 1 | + (1.53 + 1.28i)2-s + (−0.840 + 0.705i)3-s + (0.694 + 3.93i)4-s + (5.39 + 1.96i)5-s − 2.19·6-s + (14.5 + 5.30i)7-s + (−4.00 + 6.92i)8-s + (−4.47 + 25.4i)9-s + (5.73 + 9.94i)10-s + (−7.91 + 13.7i)11-s + (−3.36 − 2.82i)12-s + (1.32 + 7.52i)13-s + (15.5 + 26.8i)14-s + (−5.92 + 2.15i)15-s + (−15.0 + 5.47i)16-s + (11.1 − 63.0i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.161 + 0.135i)3-s + (0.0868 + 0.492i)4-s + (0.482 + 0.175i)5-s − 0.149·6-s + (0.786 + 0.286i)7-s + (−0.176 + 0.306i)8-s + (−0.165 + 0.940i)9-s + (0.181 + 0.314i)10-s + (−0.216 + 0.375i)11-s + (−0.0809 − 0.0679i)12-s + (0.0283 + 0.160i)13-s + (0.296 + 0.512i)14-s + (−0.101 + 0.0370i)15-s + (−0.234 + 0.0855i)16-s + (0.158 − 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.60717 + 1.23316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60717 + 1.23316i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 37 | \( 1 + (222. + 34.3i)T \) |
| good | 3 | \( 1 + (0.840 - 0.705i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-5.39 - 1.96i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-14.5 - 5.30i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (7.91 - 13.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 7.52i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-11.1 + 63.0i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-70.4 + 59.0i)T + (1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (5.71 + 9.90i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-73.3 + 127. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (35.9 + 203. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (16.9 + 29.3i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (279. - 101. i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (42.2 - 15.3i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-133. - 755. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-534. - 194. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (246. - 207. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 - 323.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-767. - 279. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (42.3 - 239. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (522. - 190. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (280. + 486. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.05355210982610612234944567742, −13.65402387681011536701303888226, −12.11055137742185583758971412838, −11.16967832565343442702247941467, −9.844951328905417375222417730606, −8.324643355205581444753479193123, −7.18143710203750482895674783991, −5.59906619177262189611742824391, −4.66998114611702198736437492565, −2.43373246222880282719517344263,
1.37979398293082327866407139135, 3.50986677451885579744954372649, 5.18992550053269926472560600458, 6.33697923256744556704845596867, 8.055784104105894967881803477143, 9.514493823734388506427561062218, 10.68426284365526485137218420016, 11.76503950429859160334504981588, 12.69476168041989930503879746190, 13.85983602277526112776319346090
