L(s) = 1 | + 1.91·2-s − 2.24·3-s + 1.67·4-s − 1.23·5-s − 4.30·6-s − 3.37·7-s − 0.631·8-s + 2.05·9-s − 2.37·10-s − 1.90·11-s − 3.75·12-s − 0.969·13-s − 6.46·14-s + 2.78·15-s − 4.55·16-s + 7.23·17-s + 3.94·18-s − 3.38·19-s − 2.06·20-s + 7.58·21-s − 3.65·22-s + 0.502·23-s + 1.42·24-s − 3.46·25-s − 1.85·26-s + 2.11·27-s − 5.63·28-s + ⋯ |
L(s) = 1 | + 1.35·2-s − 1.29·3-s + 0.835·4-s − 0.553·5-s − 1.75·6-s − 1.27·7-s − 0.223·8-s + 0.686·9-s − 0.749·10-s − 0.575·11-s − 1.08·12-s − 0.268·13-s − 1.72·14-s + 0.718·15-s − 1.13·16-s + 1.75·17-s + 0.929·18-s − 0.777·19-s − 0.462·20-s + 1.65·21-s − 0.779·22-s + 0.104·23-s + 0.290·24-s − 0.693·25-s − 0.364·26-s + 0.407·27-s − 1.06·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 + T \) |
good | 2 | \( 1 - 1.91T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 0.969T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 0.502T + 23T^{2} \) |
| 29 | \( 1 + 7.43T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 5.83T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 - 0.948T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 - 7.14T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−11.46586720143260920857737800469, −10.39878121050227667636689361145, −9.491448552674230298479859768841, −7.84399533687752038321416620869, −6.64864731095534240394190747658, −5.89672601431754534870545784780, −5.20001821010109954039075110070, −4.01448683881000773533878028347, −3.00395971167089935508228267065, 0,
3.00395971167089935508228267065, 4.01448683881000773533878028347, 5.20001821010109954039075110070, 5.89672601431754534870545784780, 6.64864731095534240394190747658, 7.84399533687752038321416620869, 9.491448552674230298479859768841, 10.39878121050227667636689361145, 11.46586720143260920857737800469