| L(s) = 1 | + (−0.884 + 0.467i)2-s + (−1.17 − 2.69i)3-s + (0.563 − 0.826i)4-s + (−2.02 − 0.949i)5-s + (2.30 + 1.83i)6-s + (1.12 + 2.39i)7-s + (−0.111 + 0.993i)8-s + (−3.85 + 4.15i)9-s + (2.23 − 0.106i)10-s + (−2.59 + 2.40i)11-s + (−2.89 − 0.547i)12-s + (−0.599 − 0.376i)13-s + (−2.11 − 1.59i)14-s + (−0.177 + 6.57i)15-s + (−0.365 − 0.930i)16-s + (2.25 − 2.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.625 + 0.330i)2-s + (−0.679 − 1.55i)3-s + (0.281 − 0.413i)4-s + (−0.905 − 0.424i)5-s + (0.939 + 0.749i)6-s + (0.423 + 0.905i)7-s + (−0.0395 + 0.351i)8-s + (−1.28 + 1.38i)9-s + (0.706 − 0.0338i)10-s + (−0.781 + 0.724i)11-s + (−0.834 − 0.157i)12-s + (−0.166 − 0.104i)13-s + (−0.563 − 0.426i)14-s + (−0.0458 + 1.69i)15-s + (−0.0913 − 0.232i)16-s + (0.546 − 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.334432 + 0.184798i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.334432 + 0.184798i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.884 - 0.467i)T \) |
| 5 | \( 1 + (2.02 + 0.949i)T \) |
| 7 | \( 1 + (-1.12 - 2.39i)T \) |
| good | 3 | \( 1 + (1.17 + 2.69i)T + (-2.04 + 2.19i)T^{2} \) |
| 11 | \( 1 + (2.59 - 2.40i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.599 + 0.376i)T + (5.64 + 11.7i)T^{2} \) |
| 17 | \( 1 + (-2.25 + 2.61i)T + (-2.53 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.33 - 4.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.61 + 2.24i)T + (3.42 - 22.7i)T^{2} \) |
| 29 | \( 1 + (4.12 - 8.56i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-7.98 + 4.60i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.946 - 5.00i)T + (-34.4 - 13.5i)T^{2} \) |
| 41 | \( 1 + (3.68 - 2.94i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-9.28 + 1.04i)T + (41.9 - 9.56i)T^{2} \) |
| 47 | \( 1 + (1.92 + 3.64i)T + (-26.4 + 38.8i)T^{2} \) |
| 53 | \( 1 + (0.0712 + 0.376i)T + (-49.3 + 19.3i)T^{2} \) |
| 59 | \( 1 + (5.53 + 0.834i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (-6.44 - 9.45i)T + (-22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (13.0 + 3.49i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.71 - 2.26i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (4.54 - 8.59i)T + (-41.1 - 60.3i)T^{2} \) |
| 79 | \( 1 + (-1.37 - 0.795i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.43 - 3.87i)T + (-36.0 + 74.7i)T^{2} \) |
| 89 | \( 1 + (-10.7 - 10.0i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-6.07 - 6.07i)T + 97iT^{2} \) |
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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
−11.34775679610309671644940062614, −10.34059490452270552277666314133, −8.958341743610443877950031229412, −8.065327177283887266548910479607, −7.61967070695322696593334320100, −6.74984144158223352971427509689, −5.64743687498938420788072539271, −4.87322267357969167861661122613, −2.59027948610529307597145910981, −1.27920850547563852719446841340,
0.33916238939648421313390720742, 3.05693518249321286305172934934, 4.00972815492834465902259515838, 4.79065201317502890598113078252, 6.09667850157798571661093159599, 7.40318591680301048478782844631, 8.230081438828879827168822974122, 9.241129303119892265573592139175, 10.32671643952455094137089107814, 10.69042647444140066884860451621
