L(s) = 1 | + (−1.40 − 0.127i)2-s + (−0.707 − 0.707i)3-s + (1.96 + 0.358i)4-s + (−0.805 + 0.805i)5-s + (0.905 + 1.08i)6-s + i·7-s + (−2.72 − 0.755i)8-s + 1.00i·9-s + (1.23 − 1.03i)10-s + (−0.326 + 0.326i)11-s + (−1.13 − 1.64i)12-s + (−3.46 − 3.46i)13-s + (0.127 − 1.40i)14-s + 1.13·15-s + (3.74 + 1.41i)16-s − 7.57·17-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0899i)2-s + (−0.408 − 0.408i)3-s + (0.983 + 0.179i)4-s + (−0.360 + 0.360i)5-s + (0.369 + 0.443i)6-s + 0.377i·7-s + (−0.963 − 0.266i)8-s + 0.333i·9-s + (0.391 − 0.326i)10-s + (−0.0985 + 0.0985i)11-s + (−0.328 − 0.474i)12-s + (−0.960 − 0.960i)13-s + (0.0340 − 0.376i)14-s + 0.294·15-s + (0.935 + 0.352i)16-s − 1.83·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0724266 + 0.183178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0724266 + 0.183178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.127i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + (0.805 - 0.805i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.326 - 0.326i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.46 + 3.46i)T + 13iT^{2} \) |
| 17 | \( 1 + 7.57T + 17T^{2} \) |
| 19 | \( 1 + (-3.34 - 3.34i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.11iT - 23T^{2} \) |
| 29 | \( 1 + (4.51 + 4.51i)T + 29iT^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 37 | \( 1 + (0.895 - 0.895i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 + (5.85 - 5.85i)T - 43iT^{2} \) |
| 47 | \( 1 - 1.34T + 47T^{2} \) |
| 53 | \( 1 + (-2.55 + 2.55i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.44 + 5.44i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.94 + 6.94i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.62 - 2.62i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.61iT - 71T^{2} \) |
| 73 | \( 1 + 13.6iT - 73T^{2} \) |
| 79 | \( 1 - 9.66T + 79T^{2} \) |
| 83 | \( 1 + (0.419 + 0.419i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.60iT - 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−11.50593152738115867072198757257, −11.17786505175443315952990209517, −9.983613312511321808125284915020, −9.229550064772625392217815559970, −7.919737617143200810040381318523, −7.42296651947993600928842392794, −6.34583315320762367206174069487, −5.23978314898066815062856544289, −3.30475171169273347687268294968, −1.94006298767625094433877408005,
0.18487873667878392732307645453, 2.28309388621000716025747327476, 4.12128092839287213798682365116, 5.24853059837348320808973232587, 6.74644972139710672946532414443, 7.25099264985880940875563983120, 8.742787349120255277724945432878, 9.149306364114256779032876215539, 10.32834056314380497971694468038, 11.03186404487497082612739355335