L(s) = 1 | − 2.18i·2-s + 1.58i·3-s − 2.78·4-s + (−0.558 + 2.16i)5-s + 3.46·6-s + 2.56i·7-s + 1.70i·8-s + 0.485·9-s + (4.73 + 1.22i)10-s − 11-s − 4.40i·12-s − 2.38i·13-s + 5.60·14-s + (−3.43 − 0.885i)15-s − 1.82·16-s + 3.12i·17-s + ⋯ |
L(s) = 1 | − 1.54i·2-s + 0.915i·3-s − 1.39·4-s + (−0.249 + 0.968i)5-s + 1.41·6-s + 0.968i·7-s + 0.603i·8-s + 0.161·9-s + (1.49 + 0.386i)10-s − 0.301·11-s − 1.27i·12-s − 0.660i·13-s + 1.49·14-s + (−0.886 − 0.228i)15-s − 0.456·16-s + 0.758i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9029704149\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9029704149\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.558 - 2.16i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 2.18iT - 2T^{2} \) |
| 3 | \( 1 - 1.58iT - 3T^{2} \) |
| 7 | \( 1 - 2.56iT - 7T^{2} \) |
| 13 | \( 1 + 2.38iT - 13T^{2} \) |
| 17 | \( 1 - 3.12iT - 17T^{2} \) |
| 23 | \( 1 - 0.112iT - 23T^{2} \) |
| 29 | \( 1 - 0.414T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 - 3.76iT - 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 9.38iT - 43T^{2} \) |
| 47 | \( 1 - 3.73iT - 47T^{2} \) |
| 53 | \( 1 + 4.22iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 0.894T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 2.50T + 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 + 4.00iT - 97T^{2} \) |
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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
−10.18534931532266641872960969972, −9.710407964699318736620967336954, −8.847637581718726180701840496843, −7.87160261438313775512410499214, −6.63067912687411277001215912635, −5.51155677093861011181066720106, −4.47606355412598488859755639753, −3.53867127923483007217238752446, −2.91876146975438249105909571141, −1.82932327020950838594195394369,
0.39881097637704929093112358341, 1.85625075504709973974900087450, 3.94031724453460629439977359914, 4.74951926590394826074993417097, 5.61732180224629669638114281358, 6.64581234464430870941855389486, 7.35597566690914492665970663693, 7.63969880703384414928419871258, 8.661015361055346609512414498914, 9.265347552122427617948978957414