| L(s) = 1 | − 0.209·2-s − 1.95·4-s − 0.591·7-s + 0.827·8-s + 0.870·11-s − 1.15·13-s + 0.123·14-s + 3.73·16-s − 4.93·17-s − 2.84·19-s − 0.182·22-s + 6.91·23-s + 0.240·26-s + 1.15·28-s + 7.48·29-s + 3.45·31-s − 2.43·32-s + 1.03·34-s − 10.1·37-s + 0.595·38-s − 9.11·41-s + 2.81·43-s − 1.70·44-s − 1.44·46-s + 6.68·47-s − 6.65·49-s + 2.25·52-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 0.978·4-s − 0.223·7-s + 0.292·8-s + 0.262·11-s − 0.319·13-s + 0.0330·14-s + 0.934·16-s − 1.19·17-s − 0.653·19-s − 0.0388·22-s + 1.44·23-s + 0.0471·26-s + 0.218·28-s + 1.39·29-s + 0.621·31-s − 0.430·32-s + 0.176·34-s − 1.67·37-s + 0.0966·38-s − 1.42·41-s + 0.429·43-s − 0.256·44-s − 0.213·46-s + 0.975·47-s − 0.950·49-s + 0.312·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.209T + 2T^{2} \) |
| 7 | \( 1 + 0.591T + 7T^{2} \) |
| 11 | \( 1 - 0.870T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 7.60T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 2.98T + 73T^{2} \) |
| 79 | \( 1 - 3.33T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 + 0.645T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.005042959925178275941585359699, −6.81969660380590598866022062642, −6.66403958357684559039674109991, −5.41128778712806888541781412712, −4.84982452023484925610807067265, −4.18330980112560714308163120358, −3.33878482625407926389778315548, −2.37005985709299735949650626536, −1.13932001034973304216091322163, 0,
1.13932001034973304216091322163, 2.37005985709299735949650626536, 3.33878482625407926389778315548, 4.18330980112560714308163120358, 4.84982452023484925610807067265, 5.41128778712806888541781412712, 6.66403958357684559039674109991, 6.81969660380590598866022062642, 8.005042959925178275941585359699
