L(s) = 1 | − 2-s + 2.77·3-s + 4-s + 5-s − 2.77·6-s + 0.220·7-s − 8-s + 4.69·9-s − 10-s − 11-s + 2.77·12-s + 4.47·13-s − 0.220·14-s + 2.77·15-s + 16-s + 7.05·17-s − 4.69·18-s + 2.09·19-s + 20-s + 0.612·21-s + 22-s + 3.13·23-s − 2.77·24-s + 25-s − 4.47·26-s + 4.70·27-s + 0.220·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.60·3-s + 0.5·4-s + 0.447·5-s − 1.13·6-s + 0.0834·7-s − 0.353·8-s + 1.56·9-s − 0.316·10-s − 0.301·11-s + 0.800·12-s + 1.23·13-s − 0.0589·14-s + 0.716·15-s + 0.250·16-s + 1.71·17-s − 1.10·18-s + 0.481·19-s + 0.223·20-s + 0.133·21-s + 0.213·22-s + 0.654·23-s − 0.566·24-s + 0.200·25-s − 0.876·26-s + 0.905·27-s + 0.0417·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.681592591\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.681592591\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 7 | \( 1 - 0.220T + 7T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 7.05T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 - 3.13T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 - 9.82T + 37T^{2} \) |
| 41 | \( 1 - 3.42T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 6.64T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 + 7.32T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.039259403784876782894167689438, −7.51801009898967390180052638217, −6.65253797665075757632795148534, −5.87928074143912605664785027992, −5.08847150031554106449526190100, −3.90727502667699446412493970844, −3.24709753432023062547014313364, −2.73190327797293789924871980526, −1.69566339488812899698794150700, −1.08296948492498728918938361386,
1.08296948492498728918938361386, 1.69566339488812899698794150700, 2.73190327797293789924871980526, 3.24709753432023062547014313364, 3.90727502667699446412493970844, 5.08847150031554106449526190100, 5.87928074143912605664785027992, 6.65253797665075757632795148534, 7.51801009898967390180052638217, 8.039259403784876782894167689438