L(s) = 1 | + (0.707 − 0.707i)2-s + (0.965 − 0.258i)3-s + (0.500 − 0.866i)6-s + (−0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + 1.00·16-s + (−0.965 − 0.258i)17-s + (0.258 − 0.965i)18-s + (0.866 + 0.5i)19-s + (−0.499 − 0.866i)21-s + (0.866 + 0.5i)24-s + (−0.500 + 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.965 − 0.258i)3-s + (0.500 − 0.866i)6-s + (−0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + 1.00·16-s + (−0.965 − 0.258i)17-s + (0.258 − 0.965i)18-s + (0.866 + 0.5i)19-s + (−0.499 − 0.866i)21-s + (0.866 + 0.5i)24-s + (−0.500 + 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.152836075\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.152836075\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.258 + 0.965i)T \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.599944183183575691215041778483, −8.723318468672411640192532865994, −7.64746129874310785373661049847, −7.40619227602422334785438608118, −6.35848522524563005834287245575, −4.86092978831002374481985380777, −4.26547356717856522543424328296, −3.38515846241032667959857457831, −2.64313276421559857091952309918, −1.57445781972414155785619589187,
1.90488381638609017449048181607, 2.93760197554213190169260213367, 3.91792195142930063995557640940, 5.03308679443230626344713403363, 5.34966177541291652188423944316, 6.72660393028981075545833182367, 7.06951340148210184225591203193, 8.198667060408948049278215685875, 8.849960674761809066764065326974, 9.731734788295441394881879335742