| L(s) = 1 | + 1.22·2-s − 0.496·4-s + 5-s − 0.451·7-s − 3.06·8-s + 1.22·10-s − 4.84·13-s − 0.553·14-s − 2.75·16-s − 0.740·17-s + 6.80·19-s − 0.496·20-s + 0.00634·23-s + 25-s − 5.93·26-s + 0.224·28-s + 0.323·29-s − 5.60·31-s + 2.73·32-s − 0.907·34-s − 0.451·35-s + 7.36·37-s + 8.34·38-s − 3.06·40-s + 10.9·41-s − 1.80·43-s + 0.00777·46-s + ⋯ |
| L(s) = 1 | + 0.866·2-s − 0.248·4-s + 0.447·5-s − 0.170·7-s − 1.08·8-s + 0.387·10-s − 1.34·13-s − 0.148·14-s − 0.689·16-s − 0.179·17-s + 1.56·19-s − 0.111·20-s + 0.00132·23-s + 0.200·25-s − 1.16·26-s + 0.0424·28-s + 0.0601·29-s − 1.00·31-s + 0.484·32-s − 0.155·34-s − 0.0763·35-s + 1.21·37-s + 1.35·38-s − 0.484·40-s + 1.71·41-s − 0.275·43-s + 0.00114·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.342982612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.342982612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 7 | \( 1 + 0.451T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 + 0.740T + 17T^{2} \) |
| 19 | \( 1 - 6.80T + 19T^{2} \) |
| 23 | \( 1 - 0.00634T + 23T^{2} \) |
| 29 | \( 1 - 0.323T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 - 7.36T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 + 8.63T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−8.017512649792536634165007458686, −7.40418903783437437327346430838, −6.56400883584150865813883587462, −5.80838044458871531234911406388, −5.17546789920061881409746373447, −4.66471719824764509179887940613, −3.72816191518387246996161789410, −2.95818910135869482254976794491, −2.18898020889964690795479091839, −0.70629315321457451829935988454,
0.70629315321457451829935988454, 2.18898020889964690795479091839, 2.95818910135869482254976794491, 3.72816191518387246996161789410, 4.66471719824764509179887940613, 5.17546789920061881409746373447, 5.80838044458871531234911406388, 6.56400883584150865813883587462, 7.40418903783437437327346430838, 8.017512649792536634165007458686
