| L(s) = 1 | + 1.28·2-s − 3.12·3-s − 0.352·4-s − 1.92·5-s − 4.00·6-s + 1.17·7-s − 3.01·8-s + 6.73·9-s − 2.46·10-s + 1.10·12-s + 4.19·13-s + 1.50·14-s + 6.00·15-s − 3.16·16-s + 4.50·17-s + 8.64·18-s + 1.94·19-s + 0.679·20-s − 3.65·21-s − 2.16·23-s + 9.42·24-s − 1.29·25-s + 5.38·26-s − 11.6·27-s − 0.413·28-s − 3.46·29-s + 7.70·30-s + ⋯ |
| L(s) = 1 | + 0.907·2-s − 1.80·3-s − 0.176·4-s − 0.860·5-s − 1.63·6-s + 0.442·7-s − 1.06·8-s + 2.24·9-s − 0.781·10-s + 0.317·12-s + 1.16·13-s + 0.401·14-s + 1.55·15-s − 0.792·16-s + 1.09·17-s + 2.03·18-s + 0.446·19-s + 0.151·20-s − 0.797·21-s − 0.450·23-s + 1.92·24-s − 0.259·25-s + 1.05·26-s − 2.24·27-s − 0.0781·28-s − 0.643·29-s + 1.40·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072322559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072322559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + 0.686T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 + 1.25T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 3.76T + 73T^{2} \) |
| 79 | \( 1 + 6.99T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.73787988061603332971441929352, −6.98400372352461467265786447004, −6.12791145059824460826908711949, −5.69907444253828053505989794506, −5.19837078038203285098731937461, −4.30081442098600001536049553215, −4.01539439389364440430006643885, −3.12416691355110211060171173371, −1.45551267735267703879181869525, −0.54399797135555712769324043849,
0.54399797135555712769324043849, 1.45551267735267703879181869525, 3.12416691355110211060171173371, 4.01539439389364440430006643885, 4.30081442098600001536049553215, 5.19837078038203285098731937461, 5.69907444253828053505989794506, 6.12791145059824460826908711949, 6.98400372352461467265786447004, 7.73787988061603332971441929352
