| L(s) = 1 | + i·2-s − 4-s − 3.57·5-s − i·8-s − 3.57i·10-s − 5.52i·11-s − 6.91i·13-s + 16-s − 2.44·17-s + 8.16i·19-s + 3.57·20-s + 5.52·22-s − 3.08i·23-s + 7.81·25-s + 6.91·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.60·5-s − 0.353i·8-s − 1.13i·10-s − 1.66i·11-s − 1.91i·13-s + 0.250·16-s − 0.594·17-s + 1.87i·19-s + 0.800·20-s + 1.17·22-s − 0.644i·23-s + 1.56·25-s + 1.35·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2440554585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2440554585\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.57T + 5T^{2} \) |
| 11 | \( 1 + 5.52iT - 11T^{2} \) |
| 13 | \( 1 + 6.91iT - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 8.16iT - 19T^{2} \) |
| 23 | \( 1 + 3.08iT - 23T^{2} \) |
| 29 | \( 1 - 5.66iT - 29T^{2} \) |
| 31 | \( 1 - 0.400iT - 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 0.0958T + 47T^{2} \) |
| 53 | \( 1 + 5.17iT - 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 - 7.06iT - 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 8.77iT - 71T^{2} \) |
| 73 | \( 1 + 0.679iT - 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + 1.64T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 11.3iT - 97T^{2} \) |
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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
−8.498793309977468897414806257656, −8.449914017536942432296442010296, −7.83893065097917760038413393967, −7.05679363899853625208377394583, −6.02935913846586736320476614082, −5.48960525458361975106486418507, −4.44332239692536639844392036236, −3.51297256755810790007856082304, −3.11618819490528918664994925440, −0.916265576618092875501930329933,
0.10558362514160779964667036656, 1.72871522719735361648903209046, 2.66445940771793337585772186843, 3.85201558737977384644231623290, 4.56090225739139885209959107607, 4.72114451060250896238421804594, 6.50310709382690956098163194673, 7.15171780041527707280569556893, 7.68505138112862196082594509311, 8.724257402600005411471446937173
