L(s) = 1 | + (0.928 − 0.371i)2-s + (−0.947 − 1.09i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−1.28 − 0.663i)6-s + (0.514 + 0.404i)7-s + (0.415 − 0.909i)8-s + (−0.155 + 1.08i)9-s + (0.981 + 0.189i)10-s + (−1.44 − 0.137i)12-s + (0.627 + 0.184i)14-s + (−0.205 − 1.43i)15-s + (0.0475 − 0.998i)16-s + (0.258 + 1.06i)18-s + (0.981 − 0.189i)20-s + (−0.0450 − 0.945i)21-s + ⋯ |
L(s) = 1 | + (0.928 − 0.371i)2-s + (−0.947 − 1.09i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−1.28 − 0.663i)6-s + (0.514 + 0.404i)7-s + (0.415 − 0.909i)8-s + (−0.155 + 1.08i)9-s + (0.981 + 0.189i)10-s + (−1.44 − 0.137i)12-s + (0.627 + 0.184i)14-s + (−0.205 − 1.43i)15-s + (0.0475 − 0.998i)16-s + (0.258 + 1.06i)18-s + (0.981 − 0.189i)20-s + (−0.0450 − 0.945i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.666525412\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666525412\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.928 + 0.371i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.235 - 0.971i)T \) |
good | 3 | \( 1 + (0.947 + 1.09i)T + (-0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.514 - 0.404i)T + (0.235 + 0.971i)T^{2} \) |
| 11 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 13 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 17 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 19 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 23 | \( 1 + (-0.327 + 0.945i)T + (-0.786 - 0.618i)T^{2} \) |
| 29 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.273 + 1.12i)T + (-0.888 - 0.458i)T^{2} \) |
| 43 | \( 1 + (0.452 - 0.132i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + (1.95 - 0.376i)T + (0.928 - 0.371i)T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (1.49 + 0.770i)T + (0.580 + 0.814i)T^{2} \) |
| 71 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 73 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 79 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 83 | \( 1 + (0.0748 - 1.57i)T + (-0.995 - 0.0950i)T^{2} \) |
| 89 | \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.940852901820660032437034695926, −8.887434184806146648281076013516, −7.57040878347983272967793000802, −6.80541264822742100311810111487, −6.29352012034516177234006582425, −5.45149985600619017292547085686, −4.93099154407968972574998119177, −3.36716615925676176740564394745, −2.20192688867633783617838210748, −1.44137481426592101864935609937,
1.80517654212942364445901424511, 3.34904259431381450044779792137, 4.49275557089891332494553602537, 4.81093829010283257932674497582, 5.76851470875529065122976116623, 6.17426704748742662351238950786, 7.40801290959085312732335132710, 8.286166903045420611672523783294, 9.453623755738673476534699776377, 10.01836188660463275746667380161