L(s) = 1 | + (−48.2 + 251. i)2-s − 8.64e3·3-s + (−6.08e4 − 2.42e4i)4-s − 4.74e5i·5-s + (4.17e5 − 2.17e6i)6-s + 5.25e6i·7-s + (9.03e6 − 1.41e7i)8-s + 3.17e7·9-s + (1.19e8 + 2.28e7i)10-s − 3.36e8·11-s + (5.26e8 + 2.09e8i)12-s + 2.31e8i·13-s + (−1.32e9 − 2.53e8i)14-s + 4.10e9i·15-s + (3.11e9 + 2.95e9i)16-s + 6.30e9·17-s + ⋯ |
L(s) = 1 | + (−0.188 + 0.982i)2-s − 1.31·3-s + (−0.928 − 0.370i)4-s − 1.21i·5-s + (0.248 − 1.29i)6-s + 0.912i·7-s + (0.538 − 0.842i)8-s + 0.737·9-s + (1.19 + 0.228i)10-s − 1.56·11-s + (1.22 + 0.488i)12-s + 0.283i·13-s + (−0.895 − 0.172i)14-s + 1.59i·15-s + (0.725 + 0.687i)16-s + 0.903·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.615854 + 0.337151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615854 + 0.337151i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (48.2 - 251. i)T \) |
good | 3 | \( 1 + 8.64e3T + 4.30e7T^{2} \) |
| 5 | \( 1 + 4.74e5iT - 1.52e11T^{2} \) |
| 7 | \( 1 - 5.25e6iT - 3.32e13T^{2} \) |
| 11 | \( 1 + 3.36e8T + 4.59e16T^{2} \) |
| 13 | \( 1 - 2.31e8iT - 6.65e17T^{2} \) |
| 17 | \( 1 - 6.30e9T + 4.86e19T^{2} \) |
| 19 | \( 1 - 2.41e10T + 2.88e20T^{2} \) |
| 23 | \( 1 - 8.96e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 + 7.07e11iT - 2.50e23T^{2} \) |
| 31 | \( 1 + 7.96e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 - 3.93e12iT - 1.23e25T^{2} \) |
| 41 | \( 1 + 2.45e12T + 6.37e25T^{2} \) |
| 43 | \( 1 - 7.81e12T + 1.36e26T^{2} \) |
| 47 | \( 1 - 1.93e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 + 1.24e14iT - 3.87e27T^{2} \) |
| 59 | \( 1 - 6.48e13T + 2.15e28T^{2} \) |
| 61 | \( 1 - 1.48e14iT - 3.67e28T^{2} \) |
| 67 | \( 1 + 6.22e13T + 1.64e29T^{2} \) |
| 71 | \( 1 - 3.09e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 - 7.83e14T + 6.50e29T^{2} \) |
| 79 | \( 1 - 7.76e14iT - 2.30e30T^{2} \) |
| 83 | \( 1 - 2.22e15T + 5.07e30T^{2} \) |
| 89 | \( 1 - 3.96e15T + 1.54e31T^{2} \) |
| 97 | \( 1 + 1.69e15T + 6.14e31T^{2} \) |
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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
−17.59910107454387504255862112603, −16.42617638439429137667119953953, −15.61241201498630618794728237845, −13.24493389731459742362005638006, −11.88652225365601641791144399694, −9.701291407538543364127549186806, −7.950472487032171658697043840584, −5.67990586044149512281247171384, −5.09835997668898320626865404970, −0.74702477533264377842603447325,
0.65478845134785127410169239897, 3.07087374864854425702341901013, 5.24375226294893415712309670074, 7.39678721911621090100976740911, 10.44353040389918361759851466350, 10.76372746917848032472304618104, 12.39437714338259075643646767653, 14.08257743373025813390724609107, 16.40613191072195418516798421005, 17.92027338783316787538927860519