| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s + 2.44·6-s − 7.54i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s − 8.65i·11-s + 3.46·12-s − 19.9·13-s − 10.6i·14-s − 3.87i·15-s + 4.00·16-s + 9.27i·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 − 1.07i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.786i·11-s + 0.288·12-s − 1.53·13-s − 0.762i·14-s − 0.258i·15-s + 0.250·16-s + 0.545i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0455 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0455 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.072447900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072447900\) |
| \(L(2)\) |
|
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 | \( 1 + (-22.9 + 1.04i)T \) |
| good | 7 | \( 1 + 7.54iT - 49T^{2} \) |
| 11 | \( 1 + 8.65iT - 121T^{2} \) |
| 13 | \( 1 + 19.9T + 169T^{2} \) |
| 17 | \( 1 - 9.27iT - 289T^{2} \) |
| 19 | \( 1 + 35.9iT - 361T^{2} \) |
| 29 | \( 1 + 27.4T + 841T^{2} \) |
| 31 | \( 1 - 22.3T + 961T^{2} \) |
| 37 | \( 1 - 42.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 64.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 58.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 29.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 95.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 106. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 80.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 103. iT - 9.40e3T^{2} \) |
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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
−10.10005589506637474459018084559, −9.205770286897613699425096075037, −8.240329091104198988187624450438, −7.27474715524885187872146114388, −6.68895323978093440554724509055, −5.19821353120655534539408811986, −4.54525807097862440827070132385, −3.46712680588601702866973959883, −2.42476128063933804952206811551, −0.78092350421102357024878559960,
2.02497581148954942660834101714, 2.71438416187035116134376063123, 3.89181657548174995614771074888, 5.04660609831630616897025780375, 5.82436087933519497086320507767, 7.10406549588423994040021077675, 7.59000437257534099177859185918, 8.793698480411661129960427264763, 9.706502466210426896622776831085, 10.34103025987451187245501221888
