| L(s) = 1 | − 3.82i·5-s − 3.07i·7-s − 0.661·11-s − 1.01·13-s + 2.00i·17-s + 1.65i·19-s − 3.45·23-s − 9.63·25-s + 3.05i·29-s − 2.68i·31-s − 11.7·35-s + 37-s + 5.81i·41-s − 1.91i·43-s − 6.92·47-s + ⋯ |
| L(s) = 1 | − 1.71i·5-s − 1.16i·7-s − 0.199·11-s − 0.281·13-s + 0.486i·17-s + 0.379i·19-s − 0.720·23-s − 1.92·25-s + 0.567i·29-s − 0.483i·31-s − 1.98·35-s + 0.164·37-s + 0.907i·41-s − 0.291i·43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3142171125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3142171125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 3.82iT - 5T^{2} \) |
| 7 | \( 1 + 3.07iT - 7T^{2} \) |
| 11 | \( 1 + 0.661T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 2.00iT - 17T^{2} \) |
| 19 | \( 1 - 1.65iT - 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 - 3.05iT - 29T^{2} \) |
| 31 | \( 1 + 2.68iT - 31T^{2} \) |
| 41 | \( 1 - 5.81iT - 41T^{2} \) |
| 43 | \( 1 + 1.91iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 3.53iT - 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 0.0232iT - 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 + 4.90iT - 79T^{2} \) |
| 83 | \( 1 + 8.07T + 83T^{2} \) |
| 89 | \( 1 - 2.13iT - 89T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−7.944844243961331011595590095119, −7.10436474226351467377435748267, −6.15915736571165698545864966164, −5.43545061193800694385647920914, −4.60564897703363935377884942137, −4.21889269162087536743254764707, −3.31558091614250811066110892795, −1.87416830786758354806403490666, −1.12590626217201932914884897591, −0.083827453320274900996555231179,
1.90624308913920837930887915911, 2.67536068840236924813522802562, 3.11703614591589750003930408126, 4.16818569254691672781135309305, 5.19271667302296914663826912584, 5.94551012543435006310691248464, 6.48848118879600003911726624265, 7.23064827448279315771214772835, 7.81748167456637425293188561621, 8.645366355790904547783411593628
