L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 1.61·6-s − 0.618·7-s + 2.23·8-s + 9-s − 0.618·12-s + 0.381·13-s + 1.00·14-s − 4.85·16-s + 0.236·17-s − 1.61·18-s − 4.23·19-s + 0.618·21-s − 2.85·23-s − 2.23·24-s − 0.618·26-s − 27-s − 0.381·28-s − 0.236·29-s − 1.14·31-s + 3.38·32-s − 0.381·34-s + 0.618·36-s + 8.94·37-s + 6.85·38-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.660·6-s − 0.233·7-s + 0.790·8-s + 0.333·9-s − 0.178·12-s + 0.105·13-s + 0.267·14-s − 1.21·16-s + 0.0572·17-s − 0.381·18-s − 0.971·19-s + 0.134·21-s − 0.595·23-s − 0.456·24-s − 0.121·26-s − 0.192·27-s − 0.0721·28-s − 0.0438·29-s − 0.205·31-s + 0.597·32-s − 0.0655·34-s + 0.103·36-s + 1.47·37-s + 1.11·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0901T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + 5.61T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 0.854T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.52772105241633030692858886589, −6.85821353079416627717073130072, −6.09746488176982449427590445114, −5.53113400640190389431023886619, −4.36427730371600209474471360991, −4.17734036854244007851074448140, −2.80574578131278843305850354054, −1.88749901354293104812017015280, −0.941104436023682331557825960416, 0,
0.941104436023682331557825960416, 1.88749901354293104812017015280, 2.80574578131278843305850354054, 4.17734036854244007851074448140, 4.36427730371600209474471360991, 5.53113400640190389431023886619, 6.09746488176982449427590445114, 6.85821353079416627717073130072, 7.52772105241633030692858886589