| L(s) = 1 | − 2.41·2-s + 3.26·3-s + 3.82·4-s − 7.88·6-s + 1.16·7-s − 4.41·8-s + 7.65·9-s − 1.93·11-s + 12.4·12-s − 3.23·13-s − 2.82·14-s + 3.00·16-s + 0.0669·17-s − 18.4·18-s − 2.33·19-s + 3.81·21-s + 4.66·22-s + 1.67·23-s − 14.4·24-s + 7.81·26-s + 15.1·27-s + 4.47·28-s − 0.300·29-s − 0.443·31-s + 1.58·32-s − 6.31·33-s − 0.161·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.88·3-s + 1.91·4-s − 3.21·6-s + 0.441·7-s − 1.56·8-s + 2.55·9-s − 0.582·11-s + 3.60·12-s − 0.897·13-s − 0.754·14-s + 0.750·16-s + 0.0162·17-s − 4.35·18-s − 0.536·19-s + 0.832·21-s + 0.995·22-s + 0.348·23-s − 2.94·24-s + 1.53·26-s + 2.92·27-s + 0.846·28-s − 0.0558·29-s − 0.0797·31-s + 0.279·32-s − 1.09·33-s − 0.0277·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.862318375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862318375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 3.26T + 3T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 1.93T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.0669T + 17T^{2} \) |
| 19 | \( 1 + 2.33T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 0.300T + 29T^{2} \) |
| 31 | \( 1 + 0.443T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 + 7.51T + 79T^{2} \) |
| 83 | \( 1 - 0.266T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.282529781333220379035025732633, −7.65541851292540610338414849516, −7.22953104817781164112477714764, −6.51452732715142375160493754775, −5.10846863926081851363527221460, −4.22137225218438366161326036538, −3.20092489712337962611170007245, −2.33329089510567088878402444825, −2.02995869104697831681587758681, −0.843145443643414308995578426128,
0.843145443643414308995578426128, 2.02995869104697831681587758681, 2.33329089510567088878402444825, 3.20092489712337962611170007245, 4.22137225218438366161326036538, 5.10846863926081851363527221460, 6.51452732715142375160493754775, 7.22953104817781164112477714764, 7.65541851292540610338414849516, 8.282529781333220379035025732633
