| L(s) = 1 | + (−2.80 − 4.85i)3-s + (−48.2 + 8.43i)7-s + (24.8 − 42.9i)9-s + (91.4 + 158. i)11-s − 206.·13-s + (−72.5 − 125. i)17-s + (127. + 73.8i)19-s + (176. + 210. i)21-s + (530. + 306. i)23-s − 731.·27-s + 63.1·29-s + (−83.2 + 48.0i)31-s + (512. − 887. i)33-s + (650. + 375. i)37-s + (579. + 1.00e3i)39-s + ⋯ |
| L(s) = 1 | + (−0.311 − 0.539i)3-s + (−0.985 + 0.172i)7-s + (0.306 − 0.530i)9-s + (0.755 + 1.30i)11-s − 1.22·13-s + (−0.250 − 0.434i)17-s + (0.354 + 0.204i)19-s + (0.399 + 0.477i)21-s + (1.00 + 0.578i)23-s − 1.00·27-s + 0.0750·29-s + (−0.0866 + 0.0500i)31-s + (0.470 − 0.814i)33-s + (0.475 + 0.274i)37-s + (0.380 + 0.659i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.155374666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155374666\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (48.2 - 8.43i)T \) |
| good | 3 | \( 1 + (2.80 + 4.85i)T + (-40.5 + 70.1i)T^{2} \) |
| 11 | \( 1 + (-91.4 - 158. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + 206.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (72.5 + 125. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-127. - 73.8i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-530. - 306. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 - 63.1T + 7.07e5T^{2} \) |
| 31 | \( 1 + (83.2 - 48.0i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-650. - 375. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 1.92e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.83e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.06e3 + 1.84e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (3.29e3 - 1.90e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-2.31e3 + 1.33e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-4.02e3 - 2.32e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-1.53e3 + 886. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 7.31e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (4.62e3 + 8.01e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-4.43e3 + 7.67e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 4.44e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (120. + 69.5i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 2.11e3T + 8.85e7T^{2} \) |
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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
−9.512741555558100139325338660013, −9.165131874865324621324179948378, −7.47845146398820059501032991159, −7.05319313330669140601197768923, −6.31135972603047544555923242152, −5.15443989685876873019391166842, −4.09280431080678688499083341658, −2.89148689966634532534135817190, −1.66655893134059784903955175677, −0.37362184960816110361083052265,
0.851579306327711210762821331526, 2.55819898459292282999754080067, 3.60579897311708397061812239958, 4.58922879166713071643097280978, 5.57698649410427798668113087922, 6.51895944677519678208218303766, 7.34049840028916880375627624078, 8.496243453946845125492421694171, 9.406865554832794034508818400249, 10.02684368612433888157295840066
