L(s) = 1 | + 2.20·2-s + (−1.59 − 0.672i)3-s + 2.87·4-s + (−1.33 + 1.79i)5-s + (−3.52 − 1.48i)6-s + 1.94·8-s + (2.09 + 2.14i)9-s + (−2.94 + 3.96i)10-s + 3.81i·11-s + (−4.59 − 1.93i)12-s + 6.50·13-s + (3.33 − 1.97i)15-s − 1.46·16-s + 2.94i·17-s + (4.63 + 4.74i)18-s + 2.03i·19-s + ⋯ |
L(s) = 1 | + 1.56·2-s + (−0.921 − 0.388i)3-s + 1.43·4-s + (−0.595 + 0.803i)5-s + (−1.43 − 0.606i)6-s + 0.687·8-s + (0.698 + 0.715i)9-s + (−0.930 + 1.25i)10-s + 1.15i·11-s + (−1.32 − 0.558i)12-s + 1.80·13-s + (0.860 − 0.509i)15-s − 0.366·16-s + 0.713i·17-s + (1.09 + 1.11i)18-s + 0.466i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11623 + 1.09573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11623 + 1.09573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.59 + 0.672i)T \) |
| 5 | \( 1 + (1.33 - 1.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 11 | \( 1 - 3.81iT - 11T^{2} \) |
| 13 | \( 1 - 6.50T + 13T^{2} \) |
| 17 | \( 1 - 2.94iT - 17T^{2} \) |
| 19 | \( 1 - 2.03iT - 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 2.25iT - 29T^{2} \) |
| 31 | \( 1 - 6.66iT - 31T^{2} \) |
| 37 | \( 1 - 3.36iT - 37T^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 + 7.03iT - 43T^{2} \) |
| 47 | \( 1 + 5.01iT - 47T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 + 13.6iT - 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 4.46T + 97T^{2} \) |
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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
−10.93383822154422195346996978985, −10.20243078169684753873470959236, −8.573732931158506831992676325675, −7.39085578740451005963563014135, −6.65269285952482635772693854377, −6.08691041186726036232524374500, −5.06844404417235034972374873041, −4.12669709546072409014480986369, −3.33011865316252625274298729232, −1.78552950119263528498553612313,
0.894763379409199846864302842472, 3.16446787402903989200857722946, 4.03514697579625991412132295654, 4.69328028638037637133445304635, 5.77995876747166557751064803975, 6.06985054164498159304404563738, 7.30810930191706173209747394489, 8.599170944962777039046427249857, 9.355460695632235591896599177010, 10.93301843894301442919077356085