L(s) = 1 | − 2.47·2-s − 3-s + 4.11·4-s + 2.47·6-s + 7-s − 5.22·8-s + 9-s − 11-s − 4.11·12-s + 5.87·13-s − 2.47·14-s + 4.70·16-s − 7.51·17-s − 2.47·18-s − 2.35·19-s − 21-s + 2.47·22-s − 6.94·23-s + 5.22·24-s − 14.5·26-s − 27-s + 4.11·28-s − 5.87·29-s − 3.66·31-s − 1.16·32-s + 33-s + 18.5·34-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.05·4-s + 1.00·6-s + 0.377·7-s − 1.84·8-s + 0.333·9-s − 0.301·11-s − 1.18·12-s + 1.62·13-s − 0.660·14-s + 1.17·16-s − 1.82·17-s − 0.582·18-s − 0.540·19-s − 0.218·21-s + 0.527·22-s − 1.44·23-s + 1.06·24-s − 2.84·26-s − 0.192·27-s + 0.777·28-s − 1.09·29-s − 0.657·31-s − 0.206·32-s + 0.174·33-s + 3.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4134335052\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4134335052\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 0.926T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.282232284561004906479498795410, −7.62867933867695748129941350212, −6.72215647399081415707391310241, −6.34550109908446233044778623116, −5.55389503331399996785643840235, −4.43176956072859133841734098150, −3.59385735051630310870946507920, −2.13035033377331299753982991456, −1.71872178744974587639122314576, −0.45260080174658908881155098716,
0.45260080174658908881155098716, 1.71872178744974587639122314576, 2.13035033377331299753982991456, 3.59385735051630310870946507920, 4.43176956072859133841734098150, 5.55389503331399996785643840235, 6.34550109908446233044778623116, 6.72215647399081415707391310241, 7.62867933867695748129941350212, 8.282232284561004906479498795410