| L(s) = 1 | − 2-s − 0.772·3-s + 4-s + 2.62·5-s + 0.772·6-s + 2.62·7-s − 8-s − 2.40·9-s − 2.62·10-s + 0.597·11-s − 0.772·12-s + 0.175·13-s − 2.62·14-s − 2.03·15-s + 16-s + 6.94·17-s + 2.40·18-s − 5.72·19-s + 2.62·20-s − 2.03·21-s − 0.597·22-s + 0.175·23-s + 0.772·24-s + 1.91·25-s − 0.175·26-s + 4.17·27-s + 2.62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.446·3-s + 0.5·4-s + 1.17·5-s + 0.315·6-s + 0.993·7-s − 0.353·8-s − 0.800·9-s − 0.831·10-s + 0.180·11-s − 0.223·12-s + 0.0486·13-s − 0.702·14-s − 0.524·15-s + 0.250·16-s + 1.68·17-s + 0.566·18-s − 1.31·19-s + 0.588·20-s − 0.443·21-s − 0.127·22-s + 0.0366·23-s + 0.157·24-s + 0.383·25-s − 0.0344·26-s + 0.803·27-s + 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.088886781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.088886781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 0.772T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 - 0.597T + 11T^{2} \) |
| 13 | \( 1 - 0.175T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 - 0.175T + 23T^{2} \) |
| 29 | \( 1 - 8.62T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 0.772T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 - 0.629T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 9.22T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−11.53886503261533368655411124214, −10.37336236014094733029761395797, −9.975743970542972682253605327197, −8.615258997960972993392054725636, −8.109069453549852273810996821938, −6.57826486554111031972700048720, −5.84217987258767075933307270228, −4.82306348846688234177878582182, −2.76298440028912646406152942860, −1.36563292974038541742435107004,
1.36563292974038541742435107004, 2.76298440028912646406152942860, 4.82306348846688234177878582182, 5.84217987258767075933307270228, 6.57826486554111031972700048720, 8.109069453549852273810996821938, 8.615258997960972993392054725636, 9.975743970542972682253605327197, 10.37336236014094733029761395797, 11.53886503261533368655411124214
