| L(s) = 1 | − 4.18·5-s − 3i·11-s − 2.44i·13-s + 1.01·17-s + 1.01i·19-s + 4.24i·23-s + 12.4·25-s − 1.24i·29-s + 5.61i·31-s + 8.24·37-s + 2.02·41-s − 8.24·43-s − 1.01·47-s − 1.24i·53-s + 12.5i·55-s + ⋯ |
| L(s) = 1 | − 1.87·5-s − 0.904i·11-s − 0.679i·13-s + 0.246·17-s + 0.232i·19-s + 0.884i·23-s + 2.49·25-s − 0.230i·29-s + 1.00i·31-s + 1.35·37-s + 0.316·41-s − 1.25·43-s − 0.147·47-s − 0.170i·53-s + 1.69i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7960261747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7960261747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.18T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 - 1.01iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 - 5.61iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 1.01T + 47T^{2} \) |
| 53 | \( 1 + 1.24iT - 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 5.91iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 + 8.36iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 3.76iT - 97T^{2} \) |
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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
−7.942774204883200604211635236839, −7.22638477089664495724373467450, −6.46239937439343388252347342477, −5.57180451810342795039875192285, −4.84116731114438628844151519518, −4.00346641668086664148750040961, −3.39786918906319396913296434696, −2.85676505142408061756831212380, −1.24320155624025672833199812804, −0.29712655509578796472748595100,
0.78848323948171955895583714197, 2.11760118990055219289141480650, 3.10284247477870233367588560940, 3.90503852488194822593673209931, 4.49652603237326107097412237543, 4.95676862255074671143287334413, 6.24955686351134562350646299651, 6.86421857193918291412183699732, 7.53791362542112883656642107573, 7.984375800897643488528037012275
