L(s) = 1 | − 3.27·3-s − 0.246·5-s + 7.73·9-s − 11-s − 3.17·13-s + 0.808·15-s − 6.49·17-s + 4.32·19-s − 3.15·23-s − 4.93·25-s − 15.5·27-s + 6.48·29-s − 1.78·31-s + 3.27·33-s + 8.38·37-s + 10.3·39-s − 0.553·41-s − 5.69·43-s − 1.90·45-s − 10.2·47-s + 21.2·51-s − 10.1·53-s + 0.246·55-s − 14.1·57-s + 6.45·59-s − 3.38·61-s + 0.782·65-s + ⋯ |
L(s) = 1 | − 1.89·3-s − 0.110·5-s + 2.57·9-s − 0.301·11-s − 0.879·13-s + 0.208·15-s − 1.57·17-s + 0.991·19-s − 0.658·23-s − 0.987·25-s − 2.98·27-s + 1.20·29-s − 0.319·31-s + 0.570·33-s + 1.37·37-s + 1.66·39-s − 0.0863·41-s − 0.867·43-s − 0.284·45-s − 1.50·47-s + 2.97·51-s − 1.39·53-s + 0.0332·55-s − 1.87·57-s + 0.840·59-s − 0.432·61-s + 0.0970·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3933348713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3933348713\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 + 0.246T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 + 0.553T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 6.45T + 59T^{2} \) |
| 61 | \( 1 + 3.38T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 0.345T + 71T^{2} \) |
| 73 | \( 1 - 2.97T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−7.62520344136535766439003064123, −6.81269848254572576902178156009, −6.41943122457210153100003101331, −5.72197783342231980717271385848, −4.91092031055895558062249943301, −4.65360225549296823819479075936, −3.75448014461418492235502258050, −2.48119710368333126414566897172, −1.50921321319647421998840315976, −0.34479783481048107003412693587,
0.34479783481048107003412693587, 1.50921321319647421998840315976, 2.48119710368333126414566897172, 3.75448014461418492235502258050, 4.65360225549296823819479075936, 4.91092031055895558062249943301, 5.72197783342231980717271385848, 6.41943122457210153100003101331, 6.81269848254572576902178156009, 7.62520344136535766439003064123