| L(s) = 1 | + 1.36·2-s + 1.04·3-s − 0.145·4-s + 0.274·5-s + 1.42·6-s + 2.68·7-s − 2.92·8-s − 1.89·9-s + 0.374·10-s + 5.28·11-s − 0.152·12-s + 6.33·13-s + 3.66·14-s + 0.288·15-s − 3.68·16-s + 0.934·17-s − 2.58·18-s − 5.23·19-s − 0.0400·20-s + 2.82·21-s + 7.19·22-s − 2.37·23-s − 3.06·24-s − 4.92·25-s + 8.63·26-s − 5.14·27-s − 0.391·28-s + ⋯ |
| L(s) = 1 | + 0.962·2-s + 0.605·3-s − 0.0727·4-s + 0.122·5-s + 0.583·6-s + 1.01·7-s − 1.03·8-s − 0.633·9-s + 0.118·10-s + 1.59·11-s − 0.0440·12-s + 1.75·13-s + 0.978·14-s + 0.0744·15-s − 0.921·16-s + 0.226·17-s − 0.609·18-s − 1.20·19-s − 0.00894·20-s + 0.615·21-s + 1.53·22-s − 0.495·23-s − 0.625·24-s − 0.984·25-s + 1.69·26-s − 0.989·27-s − 0.0739·28-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
| \(L(1)\) |
\(\approx\) |
\(2.486726736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486726736\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 - 0.274T + 5T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 - 0.934T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 0.455T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 0.382T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−11.52030218559160092001273545434, −10.99299743232143796116244601419, −9.219076524883441031209883356499, −8.847059329093849905015772421756, −7.88016478316439023130782231222, −6.25635295332770312824332172473, −5.64949151328152022013866618869, −4.10622758391882857060178131922, −3.65949999655660169416070735559, −1.86335344571238487507037983028,
1.86335344571238487507037983028, 3.65949999655660169416070735559, 4.10622758391882857060178131922, 5.64949151328152022013866618869, 6.25635295332770312824332172473, 7.88016478316439023130782231222, 8.847059329093849905015772421756, 9.219076524883441031209883356499, 10.99299743232143796116244601419, 11.52030218559160092001273545434
