| L(s) = 1 | − i·2-s − 4-s + (−1.41 − 1.73i)5-s − 1.03i·7-s + i·8-s + (−1.73 + 1.41i)10-s − 3.86·11-s − 1.03i·13-s − 1.03·14-s + 16-s + 7.46i·17-s + 19-s + (1.41 + 1.73i)20-s + 3.86i·22-s + 1.46i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.632 − 0.774i)5-s − 0.391i·7-s + 0.353i·8-s + (−0.547 + 0.447i)10-s − 1.16·11-s − 0.287i·13-s − 0.276·14-s + 0.250·16-s + 1.81i·17-s + 0.229·19-s + (0.316 + 0.387i)20-s + 0.823i·22-s + 0.305i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5974968299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5974968299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 1.03iT - 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 1.03iT - 13T^{2} \) |
| 17 | \( 1 - 7.46iT - 17T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 6.69iT - 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 - 1.79iT - 43T^{2} \) |
| 47 | \( 1 - 9.46iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.58iT - 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 4.39iT - 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−9.461873268930259659292883917092, −8.634480940995308106509103863830, −7.917040024817932683344453450353, −7.40615577002487334206473650729, −5.89035844546091371278712673716, −5.26240528908679836495069044847, −4.17313457531049384844645631165, −3.65639696927043151051583205560, −2.35858119911199858137696794656, −1.12426565605882792966256981668,
0.25485550203742488345685588379, 2.41893611294114705265966579050, 3.23866748315766386587556352949, 4.40879414550348326839609761668, 5.24615811442084443264314671865, 6.03630649485823690603410237756, 7.18011423243074513400666770484, 7.39031298305891814985455186867, 8.301248063906006985095982284111, 9.135842673625309202603153542360
