| L(s) = 1 | + (0.0523 − 0.998i)2-s + (−0.798 − 2.08i)3-s + (−0.994 − 0.104i)4-s + (1.05 + 1.97i)5-s + (−2.11 + 0.688i)6-s + (−2.58 − 0.547i)7-s + (−0.156 + 0.987i)8-s + (−1.46 + 1.31i)9-s + (2.02 − 0.946i)10-s + (−3.30 − 0.243i)11-s + (0.576 + 2.15i)12-s + (−1.09 + 0.558i)13-s + (−0.682 + 2.55i)14-s + (3.26 − 3.76i)15-s + (0.978 + 0.207i)16-s + (0.404 + 7.71i)17-s + ⋯ |
| L(s) = 1 | + (0.0370 − 0.706i)2-s + (−0.461 − 1.20i)3-s + (−0.497 − 0.0522i)4-s + (0.470 + 0.882i)5-s + (−0.865 + 0.281i)6-s + (−0.978 − 0.207i)7-s + (−0.0553 + 0.349i)8-s + (−0.487 + 0.438i)9-s + (0.640 − 0.299i)10-s + (−0.997 − 0.0735i)11-s + (0.166 + 0.621i)12-s + (−0.304 + 0.154i)13-s + (−0.182 + 0.683i)14-s + (0.843 − 0.971i)15-s + (0.244 + 0.0519i)16-s + (0.0980 + 1.87i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.335134 + 0.170612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335134 + 0.170612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0523 + 0.998i)T \) |
| 5 | \( 1 + (-1.05 - 1.97i)T \) |
| 7 | \( 1 + (2.58 + 0.547i)T \) |
| 11 | \( 1 + (3.30 + 0.243i)T \) |
| good | 3 | \( 1 + (0.798 + 2.08i)T + (-2.22 + 2.00i)T^{2} \) |
| 13 | \( 1 + (1.09 - 0.558i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.404 - 7.71i)T + (-16.9 + 1.77i)T^{2} \) |
| 19 | \( 1 + (0.691 + 6.57i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.13 - 7.95i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.0506 + 0.0697i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.46 - 6.90i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (8.35 + 3.20i)T + (27.4 + 24.7i)T^{2} \) |
| 41 | \( 1 + (-2.67 - 3.67i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.42 + 2.42i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.93 + 3.18i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (2.14 - 3.31i)T + (-21.5 - 48.4i)T^{2} \) |
| 59 | \( 1 + (0.587 - 5.58i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-2.21 + 10.4i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (3.22 - 12.0i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.42 + 10.5i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.806 + 0.652i)T + (15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-1.17 + 1.05i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.33 + 2.61i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (4.13 - 7.16i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.88 + 7.61i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.58819709450913731923377298843, −9.865474758084029718461671875334, −8.865295425956166569390851519436, −7.60621080902557622581673626182, −6.93099950814288660386937176705, −6.17058834469912265636652215907, −5.29337370240841031524245699201, −3.62209742124571393437474523257, −2.67761196208898698641351426226, −1.56544790897345067587576490854,
0.19373396412846653460902238084, 2.72723235472635174711493627914, 4.10904208527470193719695136875, 5.01101483383351438119010410156, 5.49855056023306390962365778666, 6.44254882551096687250500355825, 7.64312383963347222029829491210, 8.636101586508522902500705003784, 9.546182639436887944198424619844, 9.917139029211522439276210390865
