L(s) = 1 | + (1.19 + 0.760i)2-s + (0.844 + 1.81i)4-s + (2.23 + 0.0113i)5-s + (0.471 + 0.471i)7-s + (−0.372 + 2.80i)8-s + (2.65 + 1.71i)10-s − 0.335·11-s + (−3.50 − 3.50i)13-s + (0.203 + 0.921i)14-s + (−2.57 + 3.06i)16-s + (2.53 − 2.53i)17-s − 4.07·19-s + (1.86 + 4.06i)20-s + (−0.400 − 0.255i)22-s + (6.20 + 6.20i)23-s + ⋯ |
L(s) = 1 | + (0.843 + 0.537i)2-s + (0.422 + 0.906i)4-s + (0.999 + 0.00506i)5-s + (0.178 + 0.178i)7-s + (−0.131 + 0.991i)8-s + (0.840 + 0.541i)10-s − 0.101·11-s + (−0.971 − 0.971i)13-s + (0.0545 + 0.246i)14-s + (−0.643 + 0.765i)16-s + (0.614 − 0.614i)17-s − 0.934·19-s + (0.417 + 0.908i)20-s + (−0.0853 − 0.0544i)22-s + (1.29 + 1.29i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08093 + 1.18053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08093 + 1.18053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 - 0.760i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.0113i)T \) |
good | 7 | \( 1 + (-0.471 - 0.471i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 + (3.50 + 3.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.53 + 2.53i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + (-6.20 - 6.20i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.42iT - 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 + (2.24 - 2.24i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.80iT - 41T^{2} \) |
| 43 | \( 1 + (4.87 + 4.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.68 - 1.68i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.05 + 3.05i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 7.49iT - 61T^{2} \) |
| 67 | \( 1 + (-5.55 + 5.55i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (-5.05 + 5.05i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.85iT - 79T^{2} \) |
| 83 | \( 1 + (4.78 - 4.78i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \) |
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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
−11.80673444143981100646353441433, −10.75823524011623049562271654435, −9.746170023248717702602336089085, −8.705695321589318714815516356753, −7.57282922305018274369950413881, −6.74156170518021626149678994407, −5.43077523797506610592807500137, −5.13832729242709925244049271850, −3.40010112947897514347759916347, −2.21903052674256615618567421186,
1.66353324070089819816007902271, 2.80445627570821065585190562452, 4.35499602172483397189468031147, 5.21759323892384481257972111619, 6.32332949746831863272545244544, 7.11672969804836724485420411031, 8.784564768130301940535507701567, 9.704941933950141313021107090933, 10.52166531386425861049581403347, 11.22974605581933076355052109853