| L(s) = 1 | − 1.82·2-s + 1.33·4-s − 2.34·5-s + 1.69·7-s + 1.21·8-s + 4.28·10-s − 11-s + 4.11·13-s − 3.09·14-s − 4.88·16-s + 6.16·17-s − 19-s − 3.13·20-s + 1.82·22-s − 3.52·23-s + 0.502·25-s − 7.51·26-s + 2.26·28-s + 8.10·29-s − 2.30·31-s + 6.50·32-s − 11.2·34-s − 3.96·35-s − 6.56·37-s + 1.82·38-s − 2.84·40-s − 7.75·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.668·4-s − 1.04·5-s + 0.639·7-s + 0.428·8-s + 1.35·10-s − 0.301·11-s + 1.14·13-s − 0.826·14-s − 1.22·16-s + 1.49·17-s − 0.229·19-s − 0.701·20-s + 0.389·22-s − 0.734·23-s + 0.100·25-s − 1.47·26-s + 0.427·28-s + 1.50·29-s − 0.413·31-s + 1.14·32-s − 1.93·34-s − 0.671·35-s − 1.07·37-s + 0.296·38-s − 0.449·40-s − 1.21·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7225744816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7225744816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 + 6.56T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 4.96T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 9.37T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.057069536077474394704367821141, −8.225341212758804531127500942415, −8.027716138457059395816028662087, −7.30671570520981387280144962846, −6.26922216990592444301311564758, −5.14310535193933174197090088145, −4.19166762113412063119720813498, −3.31294132823923443089918194155, −1.77986369409756526934564283482, −0.72824538954737085834307536004,
0.72824538954737085834307536004, 1.77986369409756526934564283482, 3.31294132823923443089918194155, 4.19166762113412063119720813498, 5.14310535193933174197090088145, 6.26922216990592444301311564758, 7.30671570520981387280144962846, 8.027716138457059395816028662087, 8.225341212758804531127500942415, 9.057069536077474394704367821141
