L(s) = 1 | + 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s + 2.44·6-s + 11.4i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s + 13.8i·11-s + 3.46·12-s − 2.98·13-s + 16.1i·14-s − 3.87i·15-s + 4.00·16-s + 15.3i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.447i·5-s + 0.408·6-s + 1.63i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s + 1.26i·11-s + 0.288·12-s − 0.229·13-s + 1.15i·14-s − 0.258i·15-s + 0.250·16-s + 0.902i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.306954428\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.306954428\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (20.9 + 9.54i)T \) |
good | 7 | \( 1 - 11.4iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 + 2.98T + 169T^{2} \) |
| 17 | \( 1 - 15.3iT - 289T^{2} \) |
| 19 | \( 1 + 2.52iT - 361T^{2} \) |
| 29 | \( 1 - 11.6T + 841T^{2} \) |
| 31 | \( 1 - 2.71T + 961T^{2} \) |
| 37 | \( 1 - 31.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 52.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 59.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 81.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 42.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 58.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 16.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 60.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 44.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 37.9iT - 9.40e3T^{2} \) |
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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
−10.29632706551425032334079882125, −9.524972217004685930049625468228, −8.637807266427061997153645380035, −7.937619516093022882347302105224, −6.76020072262670137835911807859, −5.80125571419841412356635207230, −4.91329873485647169436330403890, −3.98593108821641336146643578154, −2.58459044691275424028520434787, −1.88801363970445219204389310241,
0.873166318449309637390361989273, 2.58035201809167015526775361372, 3.61050229867733080940969783708, 4.24976444546692819097509731946, 5.57335370108869571556068739358, 6.65248069783260819397474830922, 7.42409871221358735361074231868, 8.087301869694095493062593625228, 9.371884744597389297851318091759, 10.29785610442972069878567367554