L(s) = 1 | + 2.80·2-s + (−0.0756 − 2.99i)3-s + 3.88·4-s + (4.04 + 2.93i)5-s + (−0.212 − 8.42i)6-s − 2.64i·7-s − 0.321·8-s + (−8.98 + 0.454i)9-s + (11.3 + 8.24i)10-s − 3.59i·11-s + (−0.294 − 11.6i)12-s + 16.0i·13-s − 7.42i·14-s + (8.50 − 12.3i)15-s − 16.4·16-s + 26.3·17-s + ⋯ |
L(s) = 1 | + 1.40·2-s + (−0.0252 − 0.999i)3-s + 0.971·4-s + (0.809 + 0.587i)5-s + (−0.0354 − 1.40i)6-s − 0.377i·7-s − 0.0401·8-s + (−0.998 + 0.0504i)9-s + (1.13 + 0.824i)10-s − 0.327i·11-s + (−0.0245 − 0.971i)12-s + 1.23i·13-s − 0.530i·14-s + (0.566 − 0.823i)15-s − 1.02·16-s + 1.54·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.55721 - 0.794879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55721 - 0.794879i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.0756 + 2.99i)T \) |
| 5 | \( 1 + (-4.04 - 2.93i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 2 | \( 1 - 2.80T + 4T^{2} \) |
| 11 | \( 1 + 3.59iT - 121T^{2} \) |
| 13 | \( 1 - 16.0iT - 169T^{2} \) |
| 17 | \( 1 - 26.3T + 289T^{2} \) |
| 19 | \( 1 + 14.4T + 361T^{2} \) |
| 23 | \( 1 + 24.0T + 529T^{2} \) |
| 29 | \( 1 - 16.0iT - 841T^{2} \) |
| 31 | \( 1 + 11.4T + 961T^{2} \) |
| 37 | \( 1 + 37.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 4.38iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 72.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 46.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + 69.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 23.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 126. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 62.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 66.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 44.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 46.0iT - 9.40e3T^{2} \) |
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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
−13.66677446049061903227342141422, −12.57187993406834563333285275285, −11.79896571417844614371535173785, −10.60519172818382987813359467376, −9.033792947174527929961755416785, −7.33220396522021769737942949628, −6.33920632578354214265520255500, −5.49800858628641802660624408763, −3.66619666082507804235386666568, −2.13057907384420145246998601641,
2.81881745113931079619263809675, 4.26000147739579903274628247897, 5.43102470662614268774580960134, 5.96297462603788744646869741300, 8.253806078255276559164138761642, 9.554710121325639365465567055270, 10.42588830541789653164719534657, 11.92767805427996254556229492722, 12.66185687555992959620656485756, 13.70010999818978488036678437252