L(s) = 1 | − 0.107·2-s − 2.82·3-s − 1.98·4-s − 3.72·5-s + 0.303·6-s − 3.10·7-s + 0.429·8-s + 4.97·9-s + 0.401·10-s + 4.91·11-s + 5.61·12-s − 2.18·13-s + 0.334·14-s + 10.5·15-s + 3.93·16-s − 4.77·17-s − 0.535·18-s + 0.688·19-s + 7.41·20-s + 8.76·21-s − 0.528·22-s − 7.77·23-s − 1.21·24-s + 8.88·25-s + 0.235·26-s − 5.58·27-s + 6.17·28-s + ⋯ |
L(s) = 1 | − 0.0760·2-s − 1.63·3-s − 0.994·4-s − 1.66·5-s + 0.124·6-s − 1.17·7-s + 0.151·8-s + 1.65·9-s + 0.126·10-s + 1.48·11-s + 1.62·12-s − 0.606·13-s + 0.0892·14-s + 2.71·15-s + 0.982·16-s − 1.15·17-s − 0.126·18-s + 0.157·19-s + 1.65·20-s + 1.91·21-s − 0.112·22-s − 1.62·23-s − 0.247·24-s + 1.77·25-s + 0.0461·26-s − 1.07·27-s + 1.16·28-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
\(L(1)\) |
\(\approx\) |
\(0.1895690888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1895690888\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.107T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 3.72T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 0.688T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 9.63T + 47T^{2} \) |
| 53 | \( 1 - 6.04T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 0.633T + 61T^{2} \) |
| 67 | \( 1 - 0.475T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−11.61679101870428245767196961463, −10.77684695565938122613634095144, −9.685368072349372220050404025355, −8.860335175307188788442212388751, −7.47999734892711359449909975023, −6.65848771991521861122392226046, −5.62344741431607035308173301140, −4.21053403176283385368831582718, −3.93320622045546731710510617968, −0.45743924651590474213246576399,
0.45743924651590474213246576399, 3.93320622045546731710510617968, 4.21053403176283385368831582718, 5.62344741431607035308173301140, 6.65848771991521861122392226046, 7.47999734892711359449909975023, 8.860335175307188788442212388751, 9.685368072349372220050404025355, 10.77684695565938122613634095144, 11.61679101870428245767196961463