| L(s) = 1 | + 2.29·2-s + 3.25·4-s − 1.04·5-s + 1.05·7-s + 2.88·8-s − 2.40·10-s + 2.36·11-s + 2.07·13-s + 2.42·14-s + 0.107·16-s + 3.33·17-s + 1.50·19-s − 3.41·20-s + 5.42·22-s + 6.16·23-s − 3.90·25-s + 4.75·26-s + 3.44·28-s + 8.17·29-s − 0.508·31-s − 5.53·32-s + 7.65·34-s − 1.10·35-s + 2.24·37-s + 3.44·38-s − 3.02·40-s − 0.336·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.62·4-s − 0.468·5-s + 0.399·7-s + 1.02·8-s − 0.760·10-s + 0.713·11-s + 0.575·13-s + 0.647·14-s + 0.0268·16-s + 0.809·17-s + 0.345·19-s − 0.764·20-s + 1.15·22-s + 1.28·23-s − 0.780·25-s + 0.933·26-s + 0.650·28-s + 1.51·29-s − 0.0912·31-s − 0.978·32-s + 1.31·34-s − 0.187·35-s + 0.369·37-s + 0.559·38-s − 0.479·40-s − 0.0525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.317543281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.317543281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 + 0.508T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 0.336T + 41T^{2} \) |
| 43 | \( 1 + 9.87T + 43T^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + 9.64T + 53T^{2} \) |
| 59 | \( 1 + 7.95T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 0.680T + 73T^{2} \) |
| 79 | \( 1 + 0.430T + 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 + 4.64T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
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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
−9.695801439166998192034160249480, −8.677816932540404487116785824263, −7.79409946204683908815589416001, −6.85379543325865029777991992822, −6.16155904462378975401940699643, −5.22267035304835827387008134110, −4.51279667415098254744491499323, −3.63448554500407127758625014320, −2.92030950091727292995230282593, −1.40441368249473801070644812156,
1.40441368249473801070644812156, 2.92030950091727292995230282593, 3.63448554500407127758625014320, 4.51279667415098254744491499323, 5.22267035304835827387008134110, 6.16155904462378975401940699643, 6.85379543325865029777991992822, 7.79409946204683908815589416001, 8.677816932540404487116785824263, 9.695801439166998192034160249480
