L(s) = 1 | + 21.7i·2-s + 44.4·3-s − 345.·4-s + 104. i·5-s + 967. i·6-s − 454.·7-s − 4.72e3i·8-s − 207.·9-s − 2.28e3·10-s − 4.88e3·11-s − 1.53e4·12-s − 2.16e3i·13-s − 9.89e3i·14-s + 4.66e3i·15-s + 5.86e4·16-s + 3.32e4i·17-s + ⋯ |
L(s) = 1 | + 1.92i·2-s + 0.951·3-s − 2.69·4-s + 0.375i·5-s + 1.82i·6-s − 0.501·7-s − 3.26i·8-s − 0.0947·9-s − 0.721·10-s − 1.10·11-s − 2.56·12-s − 0.273i·13-s − 0.964i·14-s + 0.357i·15-s + 3.57·16-s + 1.64i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.562672 - 0.620200i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.562672 - 0.620200i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (2.98e4 - 3.06e5i)T \) |
good | 2 | \( 1 - 21.7iT - 128T^{2} \) |
| 3 | \( 1 - 44.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 104. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 454.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.88e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.16e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.32e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.45e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 6.23e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 6.11e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 3.47e4iT - 2.75e10T^{2} \) |
| 41 | \( 1 + 6.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.24e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.03e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.51e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 9.81e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 9.41e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.48e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 6.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.46e5iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 3.11e6iT - 8.07e13T^{2} \) |
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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
−15.31326710402355625342668018362, −15.00194030714111491514711994225, −13.67898415254394244741242717788, −13.01946414740029874417937010250, −10.19414390980646658360860794134, −8.808333108691584334462976538021, −7.985836181740105288603621843466, −6.74194696003570712206751812727, −5.32938763663962717975835372343, −3.40258092358853113645799915170,
0.32363272696509855186877692738, 2.29945900302183669457931221076, 3.29279628708770310615429167286, 4.94923475439064634629466285335, 8.176733730298646662784106165193, 9.221120685690701038587504999800, 10.18244963292313006487391376465, 11.54957411080722958915923783096, 12.74856185186336931782679860444, 13.56107128943276360968155803752