L(s) = 1 | + (5.12 + 5.12i)2-s + (10.3 − 10.3i)3-s + 36.5i·4-s + (−22.2 − 11.3i)5-s + 106.·6-s + (13.0 + 13.0i)7-s + (−105. + 105. i)8-s − 132. i·9-s + (−56.3 − 172. i)10-s − 135.·11-s + (378. + 378. i)12-s + (−35.4 + 35.4i)13-s + 134. i·14-s + (−347. + 113. i)15-s − 497.·16-s + (146. + 146. i)17-s + ⋯ |
L(s) = 1 | + (1.28 + 1.28i)2-s + (1.14 − 1.14i)3-s + 2.28i·4-s + (−0.891 − 0.452i)5-s + 2.94·6-s + (0.267 + 0.267i)7-s + (−1.64 + 1.64i)8-s − 1.63i·9-s + (−0.563 − 1.72i)10-s − 1.12·11-s + (2.62 + 2.62i)12-s + (−0.209 + 0.209i)13-s + 0.685i·14-s + (−1.54 + 0.505i)15-s − 1.94·16-s + (0.506 + 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.76646 + 1.28512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76646 + 1.28512i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (22.2 + 11.3i)T \) |
| 7 | \( 1 + (-13.0 - 13.0i)T \) |
good | 2 | \( 1 + (-5.12 - 5.12i)T + 16iT^{2} \) |
| 3 | \( 1 + (-10.3 + 10.3i)T - 81iT^{2} \) |
| 11 | \( 1 + 135.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (35.4 - 35.4i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-146. - 146. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 408. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-212. + 212. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 195. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 965.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-925. - 925. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 2.29e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.19e3 - 1.19e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.55e3 + 1.55e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-645. + 645. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 2.29e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.19e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (6.11e3 + 6.11e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 740.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.76e3 + 1.76e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 8.40e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-5.89e3 + 5.89e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 4.76e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-687. - 687. i)T + 8.85e7iT^{2} \) |
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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
−15.39059065056413138043054693781, −14.84059829525197620377059918099, −13.54138233588021185887294117493, −12.91975921853151144167890542894, −11.91547538747760195265017504395, −8.510560966426606922087736402470, −7.87964296141236243881487076467, −6.81142141861829439517942163454, −4.87919997230110402814278682051, −3.03799191838971872083692528809,
2.78360456896506542987146549569, 3.78852929502642859517961430017, 5.01517858982937958699127294868, 7.995651048223811574529354971223, 9.934836689424456716971172669107, 10.65136761782133181312890402627, 11.90557643836239042134820637237, 13.37443529975830038891117395674, 14.41871974633860725295671511245, 15.08774897844104608655175847464