L(s) = 1 | + (2 + 2i)2-s − 9.86i·3-s + 8i·4-s + (0.419 − 0.419i)5-s + (19.7 − 19.7i)6-s + 41.6·7-s + (−16 + 16i)8-s − 16.2·9-s + 1.67·10-s − 228. i·11-s + 78.9·12-s + (148. − 148. i)13-s + (83.2 + 83.2i)14-s + (−4.13 − 4.13i)15-s − 64·16-s + (246. − 246. i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s − 1.09i·3-s + 0.5i·4-s + (0.0167 − 0.0167i)5-s + (0.547 − 0.547i)6-s + 0.849·7-s + (−0.250 + 0.250i)8-s − 0.201·9-s + 0.0167·10-s − 1.89i·11-s + 0.547·12-s + (0.881 − 0.881i)13-s + (0.424 + 0.424i)14-s + (−0.0183 − 0.0183i)15-s − 0.250·16-s + (0.852 − 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.16550 - 0.769512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16550 - 0.769512i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 2i)T \) |
| 37 | \( 1 + (987. + 948. i)T \) |
good | 3 | \( 1 + 9.86iT - 81T^{2} \) |
| 5 | \( 1 + (-0.419 + 0.419i)T - 625iT^{2} \) |
| 7 | \( 1 - 41.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 228. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-148. + 148. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-246. + 246. i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + (494. - 494. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + (233. - 233. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-914. - 914. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-588. - 588. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 - 1.42e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-32.8 + 32.8i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.78e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.54e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (1.65e3 - 1.65e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (76.4 + 76.4i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + 2.10e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 8.19e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 1.70e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-526. + 526. i)T - 3.89e7iT^{2} \) |
| 83 | \( 1 + 7.30e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-6.32e3 - 6.32e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (4.62e3 - 4.62e3i)T - 8.85e7iT^{2} \) |
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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
−13.76476027051385513143408943279, −12.83081145888457619628864766996, −11.78772464764565147537971409848, −10.66477698639238452511111961836, −8.430922096379086115298753837740, −7.929464214883911197684451410435, −6.43495869462532990073935428566, −5.43271966806804964946016692456, −3.38812507253055875028018114537, −1.20924063068972670625322754101,
1.99075078694443529883612269892, 4.22598631141783486991282888441, 4.67615674883907343523932518764, 6.57552643943204784590412727540, 8.440582340495826977924561353118, 9.828717499515770849739251531102, 10.52720128241146384711187080197, 11.67853613909705208618738083001, 12.74309053063889393361528698756, 14.11612065891895154083036197662