L(s) = 1 | − 3.33i·5-s + (−0.662 + 2.56i)7-s + 5.03i·11-s + 6.04i·13-s − 5.20i·17-s + 4.71·19-s − 2.20i·23-s − 6.12·25-s − 7.24·29-s − 6.04·31-s + (8.54 + 2.20i)35-s − 5.12·37-s − 5.20i·41-s + 9.12i·43-s − 3.74·47-s + ⋯ |
L(s) = 1 | − 1.49i·5-s + (−0.250 + 0.968i)7-s + 1.51i·11-s + 1.67i·13-s − 1.26i·17-s + 1.08·19-s − 0.460i·23-s − 1.22·25-s − 1.34·29-s − 1.08·31-s + (1.44 + 0.373i)35-s − 0.842·37-s − 0.813i·41-s + 1.39i·43-s − 0.546·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2030972907\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2030972907\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.662 - 2.56i)T \) |
good | 5 | \( 1 + 3.33iT - 5T^{2} \) |
| 11 | \( 1 - 5.03iT - 11T^{2} \) |
| 13 | \( 1 - 6.04iT - 13T^{2} \) |
| 17 | \( 1 + 5.20iT - 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 2.20iT - 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 5.20iT - 41T^{2} \) |
| 43 | \( 1 - 9.12iT - 43T^{2} \) |
| 47 | \( 1 + 3.74T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 67 | \( 1 + 9.12iT - 67T^{2} \) |
| 71 | \( 1 + 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.12iT - 79T^{2} \) |
| 83 | \( 1 - 3.74T + 83T^{2} \) |
| 89 | \( 1 - 1.46iT - 89T^{2} \) |
| 97 | \( 1 + 6.78iT - 97T^{2} \) |
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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
−9.152412516734446807154259729503, −8.088865776378368966023127606808, −7.32511823142974336102650292855, −6.65907228774105766035087824435, −5.61287793100651082847882895571, −4.88415508718427677331741053047, −4.59064329735405233650784581617, −3.43992209043531770962411372163, −2.11418864968776214412847871988, −1.58542864419610407773391985852,
0.05634243915362390017429979890, 1.38702382144372250381460394904, 2.88507988104901410495891381937, 3.41848494475807067340142716130, 3.83431534986435532619334515767, 5.48987251579989659545140323897, 5.81211392706393435722437523118, 6.69671463499473110587124204160, 7.45657368976439008589110543194, 7.85378804107265974184982314333