L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)6-s − i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s − 25-s + (0.866 − 0.5i)26-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 − 0.866i)6-s − i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + 16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.5 + 0.866i)22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s − 25-s + (0.866 − 0.5i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4392295639\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4392295639\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.452835346745081115083108489485, −7.925133043039206103061765521945, −7.08765356738598637752699838309, −6.36620104429970610590609122435, −5.66336607753320709288782514045, −5.11123936771592119183489251876, −4.31015622373403251912049788081, −3.65369288460554169556934239256, −2.11525661783365455683824574787, −0.31312699152461624148476796347,
1.19116476975732416251713516843, 2.02035688785779943015252161692, 3.21979818444840351160258814261, 4.08278202384591875999618649238, 4.79714441016903614037997862214, 5.78589952608206587564666818171, 6.36528851443485811562798285468, 7.10644488464723404205566703346, 8.187981378069398338861974893220, 8.842697116992846785043981322275