| L(s) = 1 | + 1.27i·2-s + 0.380·4-s + (1.10 − 1.94i)5-s − 0.255i·7-s + 3.02i·8-s + (2.47 + 1.40i)10-s + 4.63·11-s + 5.02i·13-s + 0.325·14-s − 3.09·16-s − 0.336i·17-s + 2.91·19-s + (0.421 − 0.739i)20-s + 5.89i·22-s − 8.65i·23-s + ⋯ |
| L(s) = 1 | + 0.899i·2-s + 0.190·4-s + (0.495 − 0.868i)5-s − 0.0966i·7-s + 1.07i·8-s + (0.781 + 0.445i)10-s + 1.39·11-s + 1.39i·13-s + 0.0870·14-s − 0.773·16-s − 0.0815i·17-s + 0.668·19-s + (0.0941 − 0.165i)20-s + 1.25i·22-s − 1.80i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.296848145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.296848145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.10 + 1.94i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.27iT - 2T^{2} \) |
| 7 | \( 1 + 0.255iT - 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 - 5.02iT - 13T^{2} \) |
| 17 | \( 1 + 0.336iT - 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 8.65iT - 23T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 - 3.86iT - 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 - 6.98iT - 43T^{2} \) |
| 47 | \( 1 + 0.336iT - 47T^{2} \) |
| 53 | \( 1 + 6.01iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.46iT - 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 7.60iT - 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−9.398507659117832614797532454286, −8.941091145705598911256578436197, −8.201997275969050253568552655407, −7.09586911822293331608863152085, −6.53089565673122438184317085925, −5.83745982453495675837210726455, −4.79326178036366310388332699001, −4.06991917707511968415682556636, −2.39179740552220864057214526050, −1.31302907255214297481008184144,
1.16773014761747608494162207437, 2.23199338019455519856135134316, 3.31591609808190551481260963917, 3.78634453967946975046682950709, 5.48187990581555756807847670852, 6.12933772556578372561212088423, 7.13696259721001325825216665205, 7.65578579104106254764302974413, 9.211139366647384721844282148487, 9.550047141387148789610602612583
