| L(s) = 1 | + 0.554·2-s + 3.09·3-s − 1.69·4-s + 1.37·5-s + 1.71·6-s − 2.04·8-s + 6.60·9-s + 0.765·10-s + 1.80·11-s − 5.24·12-s − 4.20·13-s + 4.27·15-s + 2.24·16-s − 4.47·17-s + 3.66·18-s + 5.58·19-s − 2.33·20-s + 22-s − 0.0489·23-s − 6.34·24-s − 3.09·25-s − 2.33·26-s + 11.1·27-s − 3.15·29-s + 2.37·30-s − 6.34·31-s + 5.34·32-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 1.78·3-s − 0.846·4-s + 0.616·5-s + 0.702·6-s − 0.724·8-s + 2.20·9-s + 0.242·10-s + 0.543·11-s − 1.51·12-s − 1.16·13-s + 1.10·15-s + 0.561·16-s − 1.08·17-s + 0.863·18-s + 1.28·19-s − 0.521·20-s + 0.213·22-s − 0.0101·23-s − 1.29·24-s − 0.619·25-s − 0.457·26-s + 2.14·27-s − 0.586·29-s + 0.433·30-s − 1.14·31-s + 0.944·32-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
| \(L(1)\) |
\(\approx\) |
\(2.441176902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.441176902\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.554T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 0.0489T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + 9.44T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 - 7.30T + 59T^{2} \) |
| 61 | \( 1 + 0.954T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 + 0.273T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−11.78221235195679247677267704486, −10.04660990464125841045275335627, −9.458435488138813377901750849334, −8.944517655480454413582485881644, −7.918066961256594855003083206825, −6.93644201622684401967235448956, −5.35370927957026137542461543990, −4.22084091148520409239888904639, −3.23975346116292648871460609014, −2.00944975525541676376834252159,
2.00944975525541676376834252159, 3.23975346116292648871460609014, 4.22084091148520409239888904639, 5.35370927957026137542461543990, 6.93644201622684401967235448956, 7.918066961256594855003083206825, 8.944517655480454413582485881644, 9.458435488138813377901750849334, 10.04660990464125841045275335627, 11.78221235195679247677267704486
