| L(s) = 1 | + 0.618i·2-s + 2.23·3-s + 1.61·4-s − i·5-s + 1.38i·6-s + 4.23i·7-s + 2.23i·8-s + 2.00·9-s + 0.618·10-s − 0.236i·11-s + 3.61·12-s − 2.61·14-s − 2.23i·15-s + 1.85·16-s + 3.47·17-s + 1.23i·18-s + ⋯ |
| L(s) = 1 | + 0.437i·2-s + 1.29·3-s + 0.809·4-s − 0.447i·5-s + 0.564i·6-s + 1.60i·7-s + 0.790i·8-s + 0.666·9-s + 0.195·10-s − 0.0711i·11-s + 1.04·12-s − 0.699·14-s − 0.577i·15-s + 0.463·16-s + 0.842·17-s + 0.291i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.52688 + 1.35234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52688 + 1.35234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.618iT - 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 4.23iT - 7T^{2} \) |
| 11 | \( 1 + 0.236iT - 11T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 4.23iT - 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 11.9iT - 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 0.708iT - 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70iT - 67T^{2} \) |
| 71 | \( 1 - 6.23iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 9iT - 89T^{2} \) |
| 97 | \( 1 - 3.47iT - 97T^{2} \) |
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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
−10.05819454097039953456311381073, −9.097585350016833049898968391056, −8.657269752099306019848680704986, −7.905070590056658566257619570079, −7.10408104547085162728407925314, −5.83366021534623084359384990587, −5.34418678227535477196628536018, −3.68644505250142998103369692333, −2.60251974793724595434513290105, −1.99796744837051513733389730201,
1.37258297617396545585952263794, 2.51279461875634087467483396847, 3.58979218401086003264230983425, 3.96359422142602076216102574377, 5.83917396851247955111978295089, 6.92369324021536985361750739086, 7.66932941088087580514946389962, 8.039252372498490800099826465333, 9.564903621410016786260253556912, 10.00514495752123074450547683783
