| L(s) = 1 | + 1.54·2-s + 2.37·3-s + 0.389·4-s + 0.315·5-s + 3.66·6-s − 2.48·8-s + 2.62·9-s + 0.487·10-s − 1.80·11-s + 0.923·12-s + 5.89·13-s + 0.748·15-s − 4.62·16-s + 4.31·17-s + 4.06·18-s − 6.30·19-s + 0.122·20-s − 2.79·22-s − 6.44·23-s − 5.90·24-s − 4.90·25-s + 9.11·26-s − 0.880·27-s + 7.97·29-s + 1.15·30-s − 5.19·31-s − 2.17·32-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 1.36·3-s + 0.194·4-s + 0.141·5-s + 1.49·6-s − 0.880·8-s + 0.876·9-s + 0.154·10-s − 0.545·11-s + 0.266·12-s + 1.63·13-s + 0.193·15-s − 1.15·16-s + 1.04·17-s + 0.957·18-s − 1.44·19-s + 0.0274·20-s − 0.596·22-s − 1.34·23-s − 1.20·24-s − 0.980·25-s + 1.78·26-s − 0.169·27-s + 1.48·29-s + 0.211·30-s − 0.932·31-s − 0.383·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.007841072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.007841072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.315T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 4.10T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 9.25T + 97T^{2} \) |
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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
−11.83489027859127973120180855080, −10.57050384052962119862430808956, −9.553651218396740778054403451989, −8.496806889908795144626737526805, −8.074429265801992241275856190430, −6.44440143228858030339941611123, −5.54808152758250428122576856094, −4.07369004315205910206857214549, −3.46142965391191330713614946577, −2.20516215307468580546734423589,
2.20516215307468580546734423589, 3.46142965391191330713614946577, 4.07369004315205910206857214549, 5.54808152758250428122576856094, 6.44440143228858030339941611123, 8.074429265801992241275856190430, 8.496806889908795144626737526805, 9.553651218396740778054403451989, 10.57050384052962119862430808956, 11.83489027859127973120180855080
