L(s) = 1 | + i·2-s + i·3-s − 4-s + (2.15 + 0.594i)5-s − 6-s − 2.38i·7-s − i·8-s − 9-s + (−0.594 + 2.15i)10-s + 0.383·11-s − i·12-s − 3.74i·13-s + 2.38·14-s + (−0.594 + 2.15i)15-s + 16-s + 2.73i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.964 + 0.265i)5-s − 0.408·6-s − 0.900i·7-s − 0.353i·8-s − 0.333·9-s + (−0.187 + 0.681i)10-s + 0.115·11-s − 0.288i·12-s − 1.04i·13-s + 0.637·14-s + (−0.153 + 0.556i)15-s + 0.250·16-s + 0.664i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.893134779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.893134779\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.15 - 0.594i)T \) |
| 37 | \( 1 + iT \) |
good | 7 | \( 1 + 2.38iT - 7T^{2} \) |
| 11 | \( 1 - 0.383T + 11T^{2} \) |
| 13 | \( 1 + 3.74iT - 13T^{2} \) |
| 17 | \( 1 - 2.73iT - 17T^{2} \) |
| 19 | \( 1 - 7.29T + 19T^{2} \) |
| 23 | \( 1 - 7.68iT - 23T^{2} \) |
| 29 | \( 1 - 0.554T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 - 1.36iT - 43T^{2} \) |
| 47 | \( 1 + 1.68iT - 47T^{2} \) |
| 53 | \( 1 + 1.42iT - 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 7.67iT - 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 + 9.37T + 89T^{2} \) |
| 97 | \( 1 - 5.28iT - 97T^{2} \) |
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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
−9.850670388834408742664630550690, −9.438535835562899148633389272607, −8.251482895868565145017751934524, −7.51074681251827461903806668034, −6.64487733417274742574789985729, −5.63722689091917221324794799355, −5.16486896786135260690149123648, −3.88339285770386777788992414707, −3.02189160597283611775731040231, −1.18888356037158697117871780594,
1.09225905542472047880989066105, 2.23718676984734388456551701386, 2.93012395724391499188664157694, 4.53325994551127700662379303634, 5.36023056167957798108759456961, 6.22575774619113163296665923744, 7.07328987980868841516950670426, 8.329006063317543850204768098663, 9.017663696055232004335172669004, 9.590943993304484072922246324015