| L(s) = 1 | + (−0.284 + 1.97i)2-s + (2.88 + 6.32i)3-s + (−3.83 − 1.12i)4-s + (3.38 + 3.90i)5-s + (−13.3 + 3.91i)6-s + (−8.19 − 5.26i)7-s + (3.32 − 7.27i)8-s + (−13.9 + 16.1i)9-s + (−8.70 + 5.59i)10-s + (3.74 + 26.0i)11-s + (−3.95 − 27.5i)12-s + (0.750 − 0.482i)13-s + (12.7 − 14.7i)14-s + (−14.9 + 32.7i)15-s + (13.4 + 8.65i)16-s + (105. − 30.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.556 + 1.21i)3-s + (−0.479 − 0.140i)4-s + (0.302 + 0.349i)5-s + (−0.908 + 0.266i)6-s + (−0.442 − 0.284i)7-s + (0.146 − 0.321i)8-s + (−0.518 + 0.598i)9-s + (−0.275 + 0.176i)10-s + (0.102 + 0.713i)11-s + (−0.0952 − 0.662i)12-s + (0.0160 − 0.0102i)13-s + (0.243 − 0.281i)14-s + (−0.257 + 0.563i)15-s + (0.210 + 0.135i)16-s + (1.49 − 0.439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.699899 + 1.25137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.699899 + 1.25137i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 - 1.97i)T \) |
| 23 | \( 1 + (79.1 + 76.8i)T \) |
| good | 3 | \( 1 + (-2.88 - 6.32i)T + (-17.6 + 20.4i)T^{2} \) |
| 5 | \( 1 + (-3.38 - 3.90i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (8.19 + 5.26i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (-3.74 - 26.0i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-0.750 + 0.482i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-105. + 30.8i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-61.8 - 18.1i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-142. + 41.9i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (73.5 - 160. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (200. - 230. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (293. + 338. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-39.0 - 85.4i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 48.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + (477. + 307. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-685. + 440. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-172. + 378. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (37.2 - 259. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-73.3 + 510. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (404. + 118. i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (793. - 510. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (496. - 572. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-112. - 246. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-786. - 908. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−15.78207638791659814789102360196, −14.54296124675786703280476969659, −14.01987240792082836466484546354, −12.26918306050470369143531546786, −10.13026560982651567726034759850, −9.880945508050618814667973023130, −8.367119152506937685005581773142, −6.79700173263697067407979712447, −5.02611709889237681983017014718, −3.45313255810945433888049643065,
1.36626130318803115030487770630, 3.16996565433447257497169807306, 5.75736166711614944580353437165, 7.53292670556901757290419294756, 8.706184479140009193395687223071, 9.949198780512420764944854869283, 11.67943484292484473625907590672, 12.65541438030880595069360091834, 13.48522937469408673895622019821, 14.37671775610394582737955716227
