| L(s) = 1 | + (−0.291 + 1.37i)2-s + (−0.203 − 0.0213i)3-s + (0.0255 + 0.0113i)4-s + (−1.12 + 1.01i)5-s + (0.0887 − 0.273i)6-s + (−0.0246 − 2.64i)7-s + (−1.67 + 2.30i)8-s + (−2.89 − 0.615i)9-s + (−1.06 − 1.84i)10-s + (−0.00494 − 0.00285i)12-s + (−1.94 − 5.99i)13-s + (3.64 + 0.738i)14-s + (0.251 − 0.182i)15-s + (−2.63 − 2.93i)16-s + (2.68 − 0.569i)17-s + (1.68 − 3.79i)18-s + ⋯ |
| L(s) = 1 | + (−0.206 + 0.971i)2-s + (−0.117 − 0.0123i)3-s + (0.0127 + 0.00567i)4-s + (−0.505 + 0.454i)5-s + (0.0362 − 0.111i)6-s + (−0.00931 − 0.999i)7-s + (−0.591 + 0.814i)8-s + (−0.964 − 0.205i)9-s + (−0.337 − 0.584i)10-s + (−0.00142 − 0.000824i)12-s + (−0.540 − 1.66i)13-s + (0.973 + 0.197i)14-s + (0.0649 − 0.0471i)15-s + (−0.659 − 0.732i)16-s + (0.650 − 0.138i)17-s + (0.398 − 0.894i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.771507 - 0.207611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771507 - 0.207611i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.0246 + 2.64i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.291 - 1.37i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.203 + 0.0213i)T + (2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (1.12 - 1.01i)T + (0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.94 + 5.99i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.68 + 0.569i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-6.02 + 2.68i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.513 + 0.888i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.10 + 2.89i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.412 + 0.371i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.454 + 4.32i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (5.89 + 4.28i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 9.51iT - 43T^{2} \) |
| 47 | \( 1 + (-2.98 - 6.70i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.01 + 2.23i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.293 + 0.659i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-5.26 - 5.84i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (2.16 + 3.75i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.47 + 4.55i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.90 + 3.96i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (0.573 - 2.69i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.23 - 6.88i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.985 - 0.569i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.63 + 2.47i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.14532373808841819974462204447, −9.077915051236315295532965359760, −8.032603841911469180237177357779, −7.48598340537392074345704057914, −6.99046729432072184543423537130, −5.73251668630739470313734869096, −5.22329561389917887184976497803, −3.53091566503100287610541292590, −2.82973622612927146303619173041, −0.42112491408877863138303603716,
1.47759046471541177684106592870, 2.61505062134226875949564431641, 3.56044932345153635704214456918, 4.87341144513659493891116960809, 5.79845580075698536796604414262, 6.76256844344789190920272964958, 7.941342898814269515201691198082, 8.848974837282391835785838369931, 9.470000557021477830228950026972, 10.24609000876901267880150795434
