L(s) = 1 | + 68.3i·2-s + 2.97e3i·3-s + 2.80e4·4-s + 3.02e5·5-s − 2.03e5·6-s − 1.53e5·7-s + 4.16e6i·8-s + 5.50e6·9-s + 2.06e7i·10-s + 6.80e7i·11-s + 8.35e7i·12-s + 2.60e8·13-s − 1.04e7i·14-s + 8.99e8i·15-s + 6.36e8·16-s − 1.34e9i·17-s + ⋯ |
L(s) = 1 | + 0.377i·2-s + 0.784i·3-s + 0.857·4-s + 1.73·5-s − 0.296·6-s − 0.0702·7-s + 0.701i·8-s + 0.383·9-s + 0.654i·10-s + 1.05i·11-s + 0.672i·12-s + 1.15·13-s − 0.0265i·14-s + 1.35i·15-s + 0.592·16-s − 0.792i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(4.040506753\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.040506753\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-2.11e10 + 9.04e10i)T \) |
good | 2 | \( 1 - 68.3iT - 3.27e4T^{2} \) |
| 3 | \( 1 - 2.97e3iT - 1.43e7T^{2} \) |
| 5 | \( 1 - 3.02e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 1.53e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 6.80e7iT - 4.17e15T^{2} \) |
| 13 | \( 1 - 2.60e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.34e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 + 7.27e9iT - 1.51e19T^{2} \) |
| 23 | \( 1 + 9.77e9T + 2.66e20T^{2} \) |
| 31 | \( 1 - 4.75e10iT - 2.34e22T^{2} \) |
| 37 | \( 1 + 8.65e11iT - 3.33e23T^{2} \) |
| 41 | \( 1 - 7.42e11iT - 1.55e24T^{2} \) |
| 43 | \( 1 + 3.70e11iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 7.31e11iT - 1.20e25T^{2} \) |
| 53 | \( 1 + 1.02e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.20e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 4.06e13iT - 6.02e26T^{2} \) |
| 67 | \( 1 - 1.43e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 5.74e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.16e13iT - 8.90e27T^{2} \) |
| 79 | \( 1 - 2.57e14iT - 2.91e28T^{2} \) |
| 83 | \( 1 + 4.36e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 6.21e14iT - 1.74e29T^{2} \) |
| 97 | \( 1 + 3.36e13iT - 6.33e29T^{2} \) |
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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
−14.08488886712267155567492370944, −12.93847801962648920892062662269, −11.13324166990120114435797816101, −10.06406265306206981610173515004, −9.164353848514024964074715070062, −7.09014034536808854491863240935, −6.01495162776628532973355954054, −4.74827912868348344909304001773, −2.66332741953397711321619513854, −1.52261940718653181787281016362,
1.38284448121399971728961177075, 1.70314441614195848560957606918, 3.30666635256272918910881123282, 6.00013521683191012250523359603, 6.36699481813113820207712082928, 8.180189443699495382876897872029, 9.914417397401873690545280749659, 10.80789726591451183466909370114, 12.38376122018145357277834176519, 13.29442865172581471249335065969