| L(s) = 1 | − 0.618i·2-s + 3.23i·3-s + 1.61·4-s + 2.00·6-s + i·7-s − 2.23i·8-s − 7.47·9-s − 0.236·11-s + 5.23i·12-s − 1.23i·13-s + 0.618·14-s + 1.85·16-s + 2.47i·17-s + 4.61i·18-s + 4.47·19-s + ⋯ |
| L(s) = 1 | − 0.437i·2-s + 1.86i·3-s + 0.809·4-s + 0.816·6-s + 0.377i·7-s − 0.790i·8-s − 2.49·9-s − 0.0711·11-s + 1.51i·12-s − 0.342i·13-s + 0.165·14-s + 0.463·16-s + 0.599i·17-s + 1.08i·18-s + 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12420 + 0.694799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12420 + 0.694799i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + 0.618iT - 2T^{2} \) |
| 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 1.76iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 4.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 - 8.76iT - 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.23iT - 97T^{2} \) |
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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
−12.55833065602628834637162321434, −11.56113045395214947258880427252, −10.79593599637896004221013706162, −10.09210118858198878118867805189, −9.239963138238579781149581618329, −8.034504885805943084032925875292, −6.26153516813921453805768160005, −5.15070445306339661512659339848, −3.81330912581320927380422688031, −2.74178463747098740592672550607,
1.50130932388954037066814843703, 2.91604845681651398242151560133, 5.52387613024741033761109869961, 6.44995959987103703939611212264, 7.49034919657888033225995783149, 7.71158487448636740813452156872, 9.239063482082763984352873592714, 11.02700299587211175821092304886, 11.71273539737924117164867568784, 12.49011642332914920914563429016
