L(s) = 1 | − 1.41i·2-s + (−2.72 − 1.57i)3-s − 2.00·4-s + (−0.5 − 0.866i)5-s + (−2.22 + 3.85i)6-s + (−2.5 + 0.866i)7-s + 2.82i·8-s + (3.44 + 5.97i)9-s + (−1.22 + 0.707i)10-s + (2.44 − 4.24i)11-s + (5.44 + 3.14i)12-s + 0.449·13-s + (1.22 + 3.53i)14-s + 3.14i·15-s + 4.00·16-s + (−1.77 − 1.02i)17-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (−1.57 − 0.908i)3-s − 1.00·4-s + (−0.223 − 0.387i)5-s + (−0.908 + 1.57i)6-s + (−0.944 + 0.327i)7-s + 1.00i·8-s + (1.14 + 1.99i)9-s + (−0.387 + 0.223i)10-s + (0.738 − 1.27i)11-s + (1.57 + 0.908i)12-s + 0.124·13-s + (0.327 + 0.944i)14-s + 0.812i·15-s + 1.00·16-s + (−0.430 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 3 | \( 1 + (2.72 + 1.57i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 + (1.77 + 1.02i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.67 - 2.12i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.949 - 0.548i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10.0iT - 29T^{2} \) |
| 31 | \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 1.73i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.36iT - 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.22 + 2.43i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.12 + 5.26i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.17 + 12.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.17 + 2.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + (2.32 + 1.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.674 + 0.389i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.2iT - 83T^{2} \) |
| 89 | \( 1 + (0.398 - 0.230i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.02iT - 97T^{2} \) |
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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
−11.09874401485015249161189485020, −10.68078958275007542786222917939, −9.281772559600105749965630978990, −8.340101778864542092392593481380, −6.79105016565175011509637243879, −5.94785832514809899906644857258, −4.97530387522793433700803239173, −3.45844640485722810805123464210, −1.48247515813452362094316225641, 0,
4.00471873064514068208803733396, 4.48135591222057677570545678753, 5.94408069937196695075850119713, 6.50948641537538032316247065784, 7.34801701601554728053257820494, 9.125906448384533328275340505616, 9.880706078934278131262986246366, 10.53899096652450263562332062058, 11.72202730138941023870165872258