| L(s) = 1 | + 3.14·3-s − 3.50·5-s + 7-s + 6.91·9-s − 4.85·11-s − 11.0·15-s − 1.74·17-s + 1.73·19-s + 3.14·21-s − 8.28·23-s + 7.27·25-s + 12.3·27-s + 8.38·29-s + 0.604·31-s − 15.2·33-s − 3.50·35-s − 10.7·37-s + 5.59·41-s + 5.19·43-s − 24.2·45-s + 10.0·47-s + 49-s − 5.50·51-s + 0.668·53-s + 17.0·55-s + 5.45·57-s + 3.62·59-s + ⋯ |
| L(s) = 1 | + 1.81·3-s − 1.56·5-s + 0.377·7-s + 2.30·9-s − 1.46·11-s − 2.84·15-s − 0.424·17-s + 0.397·19-s + 0.687·21-s − 1.72·23-s + 1.45·25-s + 2.37·27-s + 1.55·29-s + 0.108·31-s − 2.66·33-s − 0.592·35-s − 1.76·37-s + 0.874·41-s + 0.792·43-s − 3.61·45-s + 1.46·47-s + 0.142·49-s − 0.771·51-s + 0.0918·53-s + 2.29·55-s + 0.723·57-s + 0.472·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.756346378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.756346378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 0.604T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.668T + 53T^{2} \) |
| 59 | \( 1 - 3.62T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 - 4.21T + 97T^{2} \) |
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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
−7.909300659239265671520857968638, −7.37685566789312370809030695485, −6.75428064407373882920408085556, −5.44359410834077792027670703606, −4.59293923274714419798810569703, −4.01382832252364584950409874626, −3.47166484894435208224045761180, −2.62522475074821575900764056147, −2.12348788764605515880780588915, −0.70243756901828416037622546321,
0.70243756901828416037622546321, 2.12348788764605515880780588915, 2.62522475074821575900764056147, 3.47166484894435208224045761180, 4.01382832252364584950409874626, 4.59293923274714419798810569703, 5.44359410834077792027670703606, 6.75428064407373882920408085556, 7.37685566789312370809030695485, 7.909300659239265671520857968638
