| L(s) = 1 | + 2.45·2-s − 0.386·3-s + 4.00·4-s + 3.21·5-s − 0.947·6-s − 4.43·7-s + 4.92·8-s − 2.85·9-s + 7.89·10-s + 6.20·11-s − 1.54·12-s − 2.07·13-s − 10.8·14-s − 1.24·15-s + 4.05·16-s − 7.12·17-s − 6.98·18-s + 0.373·19-s + 12.9·20-s + 1.71·21-s + 15.2·22-s + 1.86·23-s − 1.90·24-s + 5.36·25-s − 5.07·26-s + 2.26·27-s − 17.7·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.223·3-s + 2.00·4-s + 1.43·5-s − 0.386·6-s − 1.67·7-s + 1.74·8-s − 0.950·9-s + 2.49·10-s + 1.86·11-s − 0.447·12-s − 0.574·13-s − 2.90·14-s − 0.321·15-s + 1.01·16-s − 1.72·17-s − 1.64·18-s + 0.0856·19-s + 2.88·20-s + 0.373·21-s + 3.24·22-s + 0.389·23-s − 0.388·24-s + 1.07·25-s − 0.995·26-s + 0.435·27-s − 3.35·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.297409425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.297409425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 + 0.386T + 3T^{2} \) |
| 5 | \( 1 - 3.21T + 5T^{2} \) |
| 7 | \( 1 + 4.43T + 7T^{2} \) |
| 11 | \( 1 - 6.20T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 0.373T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 + 4.79T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 - 0.545T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 + 9.87T + 53T^{2} \) |
| 59 | \( 1 - 3.72T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 - 5.99T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 0.177T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−11.76625531407701143381939221181, −10.94021353294132493774443065938, −9.558287812004139328617173425304, −9.145387871274404661684748584404, −6.72775037936832327400596007729, −6.41137848775485270498608426566, −5.78366851155806490824181125607, −4.51041123493108903557239626631, −3.27738046784041945184851594901, −2.26897193481416708880051173246,
2.26897193481416708880051173246, 3.27738046784041945184851594901, 4.51041123493108903557239626631, 5.78366851155806490824181125607, 6.41137848775485270498608426566, 6.72775037936832327400596007729, 9.145387871274404661684748584404, 9.558287812004139328617173425304, 10.94021353294132493774443065938, 11.76625531407701143381939221181
