L(s) = 1 | + (−2.67 − 1.54i)2-s + (32.7 + 56.7i)3-s + (−59.2 − 102. i)4-s + (−63.8 + 36.8i)5-s − 202. i·6-s + (457. + 791. i)7-s + 761. i·8-s + (−1.05e3 + 1.81e3i)9-s + 227.·10-s − 5.54e3·11-s + (3.87e3 − 6.71e3i)12-s + (−810. + 468. i)13-s − 2.82e3i·14-s + (−4.17e3 − 2.41e3i)15-s + (−6.40e3 + 1.10e4i)16-s + (−1.32e4 − 7.66e3i)17-s + ⋯ |
L(s) = 1 | + (−0.236 − 0.136i)2-s + (0.700 + 1.21i)3-s + (−0.462 − 0.801i)4-s + (−0.228 + 0.131i)5-s − 0.382i·6-s + (0.503 + 0.872i)7-s + 0.526i·8-s + (−0.480 + 0.831i)9-s + 0.0720·10-s − 1.25·11-s + (0.647 − 1.12i)12-s + (−0.102 + 0.0590i)13-s − 0.275i·14-s + (−0.319 − 0.184i)15-s + (−0.390 + 0.676i)16-s + (−0.655 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.230491 + 0.888606i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.230491 + 0.888606i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-2.07e5 + 2.28e5i)T \) |
good | 2 | \( 1 + (2.67 + 1.54i)T + (64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-32.7 - 56.7i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (63.8 - 36.8i)T + (3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-457. - 791. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + 5.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (810. - 468. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.32e4 + 7.66e3i)T + (2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.93e4 - 2.27e4i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 - 1.53e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 3.34e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.02e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (-1.35e5 - 2.34e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 - 8.11e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.65e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-6.77e5 + 1.17e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.72e6 - 9.94e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.28e6 - 7.39e5i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.09e6 - 1.89e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.50e5 - 2.60e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 3.94e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-2.81e6 + 1.62e6i)T + (9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-3.52e5 + 6.10e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.88e6 + 5.12e6i)T + (2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.06e7iT - 8.07e13T^{2} \) |
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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
−15.07988026290695332618094491022, −14.70057820571631074012079201087, −13.18566732805317851014382751074, −11.23418557247739496348551492826, −10.22416638204318760453171959156, −9.177782388980744682670497968346, −8.194586563671214810112866261024, −5.56786761989605095146971036389, −4.35012598507676964112724813759, −2.37404437253418149542225557938,
0.39046543192211702788823494161, 2.44012728920569295500111825849, 4.35481972983399353692589634459, 6.95778562468016785773207904068, 7.919490942628339129320090997133, 8.625373466752185597130069080385, 10.62315049290465873880629481438, 12.37829227699574929705216709429, 13.21699876939810386432710822403, 13.91984676383386730394935088388