L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (−1.52 + 1.63i)5-s + 1.00·6-s + (2.68 + 2.68i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (0.0835 + 2.23i)10-s + 0.403i·11-s + (0.707 − 0.707i)12-s + (−2.12 − 2.12i)13-s + 3.80·14-s + (−2.23 + 0.0835i)15-s − 1.00·16-s + (4.95 + 4.95i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.680 + 0.733i)5-s + 0.408·6-s + (1.01 + 1.01i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.0264 + 0.706i)10-s + 0.121i·11-s + (0.204 − 0.204i)12-s + (−0.589 − 0.589i)13-s + 1.01·14-s + (−0.576 + 0.0215i)15-s − 0.250·16-s + (1.20 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90600 + 0.818438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90600 + 0.818438i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.52 - 1.63i)T \) |
| 23 | \( 1 + (-2.47 - 4.10i)T \) |
good | 7 | \( 1 + (-2.68 - 2.68i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.403iT - 11T^{2} \) |
| 13 | \( 1 + (2.12 + 2.12i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.95 - 4.95i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.20T + 19T^{2} \) |
| 29 | \( 1 - 9.42iT - 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 + (2.17 + 2.17i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + (-2.56 + 2.56i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.64 + 8.64i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.47 + 1.47i)T - 53iT^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 7.51iT - 61T^{2} \) |
| 67 | \( 1 + (4.60 + 4.60i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 + (-1.06 - 1.06i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.383T + 79T^{2} \) |
| 83 | \( 1 + (12.3 - 12.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 + (8.25 + 8.25i)T + 97iT^{2} \) |
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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
−10.72490179982943659582744578873, −9.986775166845608777400953655185, −8.764178829028595842068669360576, −8.128701230413659805927393654044, −7.16734038068614327182407487313, −5.79961034406316903213444349039, −5.01924627876372464460319454069, −3.91055393996976206163748864250, −2.99799024109340954333808131382, −1.91016956777339157585183119255,
0.957120456538859579478742138989, 2.68591144745999039650286510399, 4.35690398182771306463975568443, 4.43678801745782015288341827507, 5.83608000174336840793513339085, 7.17592705602521738335153604252, 7.60531117337839751487965558496, 8.345750197313689364596924620471, 9.209838800886898421685303164537, 10.41445094783149815817291116445