L(s) = 1 | + 3.50·2-s + (2.88 − 0.822i)3-s + 8.29·4-s + (10.1 − 2.88i)6-s + 2.64i·7-s + 15.0·8-s + (7.64 − 4.74i)9-s − 7.01i·11-s + (23.9 − 6.82i)12-s + 11.6i·13-s + 9.27i·14-s + 19.5·16-s − 4.52·17-s + (26.8 − 16.6i)18-s − 16.2·19-s + ⋯ |
L(s) = 1 | + 1.75·2-s + (0.961 − 0.274i)3-s + 2.07·4-s + (1.68 − 0.480i)6-s + 0.377i·7-s + 1.88·8-s + (0.849 − 0.527i)9-s − 0.637i·11-s + (1.99 − 0.568i)12-s + 0.895i·13-s + 0.662i·14-s + 1.22·16-s − 0.266·17-s + (1.48 − 0.924i)18-s − 0.854·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(6.577007349\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.577007349\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.88 + 0.822i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 2 | \( 1 - 3.50T + 4T^{2} \) |
| 11 | \( 1 + 7.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6iT - 169T^{2} \) |
| 17 | \( 1 + 4.52T + 289T^{2} \) |
| 19 | \( 1 + 16.2T + 361T^{2} \) |
| 23 | \( 1 + 25.5T + 529T^{2} \) |
| 29 | \( 1 + 9.49iT - 841T^{2} \) |
| 31 | \( 1 - 28.7T + 961T^{2} \) |
| 37 | \( 1 + 33.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 67.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 33.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 92.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 15.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 23.1iT - 9.40e3T^{2} \) |
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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
−11.01764801593169216442569315981, −9.756881631522756612302852332176, −8.692332358840551650563794259550, −7.76682000891911080658484193614, −6.57844418286249444349198339984, −6.06865492446162727855660998710, −4.64060178746772001027237691033, −3.91191534894362798689482651886, −2.82933241469229204026137297807, −1.91533682047349905411500285524,
1.99517781029072979022218345859, 3.00389501577775449227561414677, 4.03165429706993927815201551912, 4.65855108387850704520744380967, 5.81601820711913577837744587248, 6.91223462741953654407127595359, 7.72287694054385483488468212175, 8.786171821851555102855031381019, 10.16345578403717864350414893598, 10.63829335365080442465274327272