| L(s) = 1 | + 2.41·2-s + 3-s + 3.84·4-s + 1.98·5-s + 2.41·6-s + 7-s + 4.45·8-s + 9-s + 4.79·10-s + 5.56·11-s + 3.84·12-s + 2.41·14-s + 1.98·15-s + 3.07·16-s − 5.66·17-s + 2.41·18-s + 2.77·19-s + 7.61·20-s + 21-s + 13.4·22-s − 2.20·23-s + 4.45·24-s − 1.06·25-s + 27-s + 3.84·28-s − 5.79·29-s + 4.79·30-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.92·4-s + 0.886·5-s + 0.986·6-s + 0.377·7-s + 1.57·8-s + 0.333·9-s + 1.51·10-s + 1.67·11-s + 1.10·12-s + 0.645·14-s + 0.511·15-s + 0.768·16-s − 1.37·17-s + 0.569·18-s + 0.636·19-s + 1.70·20-s + 0.218·21-s + 2.86·22-s − 0.458·23-s + 0.908·24-s − 0.213·25-s + 0.192·27-s + 0.726·28-s − 1.07·29-s + 0.874·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.356184250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.356184250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 5.79T + 29T^{2} \) |
| 31 | \( 1 - 0.187T + 31T^{2} \) |
| 37 | \( 1 + 5.00T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 6.93T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 + 6.18T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 0.650T + 97T^{2} \) |
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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
−8.639233816492913663188034158910, −7.52147120444558602408232730541, −6.70419984602748174373350402133, −6.27191532748081468495878353930, −5.46168787962201135559626364856, −4.63425989740368616012306674551, −3.95791517359852156932524483580, −3.27582443016086621972519527039, −2.13085911965305307730142859891, −1.65351213767291866126082827211,
1.65351213767291866126082827211, 2.13085911965305307730142859891, 3.27582443016086621972519527039, 3.95791517359852156932524483580, 4.63425989740368616012306674551, 5.46168787962201135559626364856, 6.27191532748081468495878353930, 6.70419984602748174373350402133, 7.52147120444558602408232730541, 8.639233816492913663188034158910
