| L(s) = 1 | − 1.79i·3-s − 4.79i·7-s − 0.208·9-s + 11-s − i·13-s + 3.79i·17-s − 2.58·19-s − 8.58·21-s − 0.791i·23-s − 5.00i·27-s − 2.20·29-s − 0.582·31-s − 1.79i·33-s − 6.58i·37-s − 1.79·39-s + ⋯ |
| L(s) = 1 | − 1.03i·3-s − 1.81i·7-s − 0.0695·9-s + 0.301·11-s − 0.277i·13-s + 0.919i·17-s − 0.592·19-s − 1.87·21-s − 0.164i·23-s − 0.962i·27-s − 0.410·29-s − 0.104·31-s − 0.311i·33-s − 1.08i·37-s − 0.286·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180815226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180815226\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.79iT - 3T^{2} \) |
| 7 | \( 1 + 4.79iT - 7T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3.79iT - 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 0.791iT - 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 + 0.582T + 31T^{2} \) |
| 37 | \( 1 + 6.58iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 2.37iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 + 3.20iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−7.85680858365828173266983597924, −7.15035279647349684595859278470, −6.69493040189172339087110495357, −6.04147367387263024640823581018, −4.86301513005107134008589959840, −4.03745132899992785062877227856, −3.46487061229670831222301342209, −2.04586758329239130557493393719, −1.30514125433413281443499336127, −0.32653171371801238870142165391,
1.67458227338111246993113538677, 2.63469734103097501913416967577, 3.42232588254245779406412729190, 4.38482626561072370250400755569, 5.08403169885113036683159514632, 5.63662770458009889920042587895, 6.49304276576406467385093610328, 7.27088474486354461393188639353, 8.441663858951828520304649981694, 8.790409604932650431007111243200
