L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.436 + 0.436i)3-s + 1.00i·4-s + (0.925 − 2.03i)5-s − 0.617i·6-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s − 2.61i·9-s + (−2.09 + 0.784i)10-s + (−0.0914 − 3.31i)11-s + (−0.436 + 0.436i)12-s + (−2.24 + 2.24i)13-s + 1.00i·14-s + (1.29 − 0.484i)15-s − 1.00·16-s + (−0.882 − 0.882i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.251 + 0.251i)3-s + 0.500i·4-s + (0.413 − 0.910i)5-s − 0.251i·6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s − 0.873i·9-s + (−0.662 + 0.248i)10-s + (−0.0275 − 0.999i)11-s + (−0.125 + 0.125i)12-s + (−0.622 + 0.622i)13-s + 0.267i·14-s + (0.333 − 0.125i)15-s − 0.250·16-s + (−0.214 − 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.285944 - 0.895670i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.285944 - 0.895670i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.925 + 2.03i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.0914 + 3.31i)T \) |
good | 3 | \( 1 + (-0.436 - 0.436i)T + 3iT^{2} \) |
| 13 | \( 1 + (2.24 - 2.24i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.882 + 0.882i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 + (-4.55 - 4.55i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.41iT - 41T^{2} \) |
| 43 | \( 1 + (-6.96 + 6.96i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.01 + 6.01i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.55 + 8.55i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.83iT - 59T^{2} \) |
| 61 | \( 1 + 6.74iT - 61T^{2} \) |
| 67 | \( 1 + (-1.48 + 1.48i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 + (-2.44 + 2.44i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 + (-3.86 + 3.86i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.55iT - 89T^{2} \) |
| 97 | \( 1 + (2.55 - 2.55i)T - 97iT^{2} \) |
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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
−9.752763295662101401954682636418, −9.075643183266263328572155903133, −8.773632765418935258119947499232, −7.54909269504022754863479217243, −6.51992817350722217511104210927, −5.46115691986179757790158942496, −4.25047137167860445215925172912, −3.38863291699802300835946805031, −1.99628322970427960329109520494, −0.51654518369685464577297395412,
1.99125134711577372536026935976, 2.75565493089533120414475914660, 4.47162186110969215371195405663, 5.54106982706196784390402984535, 6.50511643036741678611626680967, 7.29123254004813225163562823115, 7.87969723534569932098880107090, 9.002520745769902445650322628980, 9.723009799939515162297036864094, 10.71673892432568312089597542940