| L(s) = 1 | + (1.92 − 1.13i)5-s + 4.74i·7-s − 4.48·11-s − 0.843i·13-s − 5.52i·17-s − 19-s − 0.779i·23-s + (2.40 − 4.38i)25-s − 10.6·29-s − 8.65·31-s + (5.40 + 9.12i)35-s + 1.62i·37-s − 4.73·41-s + 9.67i·43-s − 3.18i·47-s + ⋯ |
| L(s) = 1 | + (0.860 − 0.509i)5-s + 1.79i·7-s − 1.35·11-s − 0.233i·13-s − 1.33i·17-s − 0.229·19-s − 0.162i·23-s + (0.480 − 0.876i)25-s − 1.97·29-s − 1.55·31-s + (0.913 + 1.54i)35-s + 0.266i·37-s − 0.739·41-s + 1.47i·43-s − 0.464i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2986807720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2986807720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.92 + 1.13i)T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.74iT - 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 + 0.843iT - 13T^{2} \) |
| 17 | \( 1 + 5.52iT - 17T^{2} \) |
| 23 | \( 1 + 0.779iT - 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 - 1.62iT - 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 9.67iT - 43T^{2} \) |
| 47 | \( 1 + 3.18iT - 47T^{2} \) |
| 53 | \( 1 + 6.17iT - 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 + 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 0.303T + 71T^{2} \) |
| 73 | \( 1 + 10.0iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 0.779iT - 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 + 6.17iT - 97T^{2} \) |
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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
−8.388971431692338716390088997157, −7.70869840170698006407459884200, −6.69676827428618250797864102844, −5.70136932790392759488592445623, −5.37661313127901911406827675220, −4.86183893189053090612821986192, −3.30279008498296912165857535970, −2.43400917714125887935456497791, −1.89616114272591128344172978176, −0.079860851353144921792423181940,
1.47312030256412837262591594197, 2.33234813910457741164085652460, 3.60629833252556377954852454571, 4.04590460546789888029631231058, 5.32969739854008232653133319891, 5.77773861383525757460768708871, 6.96060598068256305907792141975, 7.22537809846921495970509304347, 8.024510592851229275002508743716, 8.926986789995764137064940230465
