| L(s) = 1 | + (−2.73 − 0.732i)2-s + (−14.4 − 8.33i)3-s + (6.92 + 4i)4-s + (44.0 − 11.7i)5-s + (33.3 + 33.3i)6-s + (20.5 − 35.5i)7-s + (−15.9 − 16i)8-s + (98.3 + 170. i)9-s − 128.·10-s − 157. i·11-s + (−66.6 − 115. i)12-s + (7.07 − 1.89i)13-s + (−82.0 + 82.0i)14-s + (−733. − 196. i)15-s + (31.9 + 55.4i)16-s + (58.3 − 217. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (−1.60 − 0.925i)3-s + (0.433 + 0.250i)4-s + (1.76 − 0.471i)5-s + (0.925 + 0.925i)6-s + (0.418 − 0.724i)7-s + (−0.249 − 0.250i)8-s + (1.21 + 2.10i)9-s − 1.28·10-s − 1.29i·11-s + (−0.462 − 0.801i)12-s + (0.0418 − 0.0112i)13-s + (−0.418 + 0.418i)14-s + (−3.26 − 0.873i)15-s + (0.124 + 0.216i)16-s + (0.202 − 0.753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.563i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.825 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.251968 - 0.815898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251968 - 0.815898i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 37 | \( 1 + (843. + 1.07e3i)T \) |
| good | 3 | \( 1 + (14.4 + 8.33i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-44.0 + 11.7i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-20.5 + 35.5i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 157. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-7.07 + 1.89i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-58.3 + 217. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (360. - 96.6i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-180. - 180. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (619. - 619. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (36.7 - 36.7i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.94e3 + 1.12e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (951. + 951. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 3.04e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-650. - 1.12e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.03e3 - 3.84e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.27e3 + 4.75e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (761. + 439. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.46e3 - 4.26e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 2.45e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-956. + 256. i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (1.57e3 + 2.73e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.43e4 - 3.85e3i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-4.40e3 - 4.40e3i)T + 8.85e7iT^{2} \) |
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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
−13.27217611201240348348761514534, −12.20521801787477487881961584559, −10.92550028742782456122100447986, −10.40818080013136707087753740246, −8.871703446472425461478158504625, −7.23814919538914123978641264913, −6.11251834372359909919993872253, −5.26914183091839161825193595406, −1.77551273782677156693434504141, −0.68927313767779874094054635048,
1.86787124511458639618117340273, 4.89536157368717415478019054942, 5.91047470507046465159845663463, 6.70612596172693892883070596983, 9.076957201201324640297586898908, 10.06316787360920466171049019737, 10.52165781604175868843186487044, 11.74263538000207768066777845477, 12.92168387723750176206054503058, 14.83775288403416915376313689160
