L(s) = 1 | + 4·2-s + (−4.24 + 15.0i)3-s + 16·4-s − 2.81i·5-s + (−16.9 + 60.0i)6-s − 187.·7-s + 64·8-s + (−207. − 127. i)9-s − 11.2i·10-s − 666.·11-s + (−67.8 + 240. i)12-s + 37.6i·13-s − 748.·14-s + (42.2 + 11.9i)15-s + 256·16-s − 547. i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.272 + 0.962i)3-s + 0.5·4-s − 0.0503i·5-s + (−0.192 + 0.680i)6-s − 1.44·7-s + 0.353·8-s + (−0.852 − 0.523i)9-s − 0.0355i·10-s − 1.66·11-s + (−0.136 + 0.481i)12-s + 0.0617i·13-s − 1.02·14-s + (0.0484 + 0.0136i)15-s + 0.250·16-s − 0.459i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.694370505\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.694370505\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + (4.24 - 15.0i)T \) |
| 59 | \( 1 + (113. + 2.67e4i)T \) |
good | 5 | \( 1 + 2.81iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 187.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 37.6iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 547. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.84e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.40e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 2.00e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 3.75e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.08e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.40e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.46e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5iT - 8.58e9T^{2} \) |
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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
−10.65075327482537544512614224997, −9.804943054220793393221333221056, −9.059184212034242798937096302016, −7.57407446814233697890199888065, −6.53057565509051089576057147474, −5.42002145122020366815474251422, −4.84053896085475641978988166804, −3.27242469034957306400537992013, −2.90834348509663208114643629315, −0.44476823591658338813868906647,
0.876275269255389010630206746881, 2.61585422565461236301251990533, 3.20536249626200166517841507829, 5.04812204603048761143333941783, 5.81110153595159293882820582746, 6.83432043686180541369423147385, 7.47209538081883739753304941924, 8.647890757383477239559520354645, 10.02052913534595434776476119395, 10.79260153481416674067356424516