L(s) = 1 | − 1.30·2-s − 0.302·4-s − 0.697·7-s + 3·8-s + 11-s − 5·13-s + 0.908·14-s − 3.30·16-s + 6.90·17-s − 19-s − 1.30·22-s − 7.30·23-s + 6.51·26-s + 0.211·28-s − 0.908·29-s + 10.2·31-s − 1.69·32-s − 9·34-s − 2.39·37-s + 1.30·38-s + 5.60·41-s − 7.21·43-s − 0.302·44-s + 9.51·46-s − 3·47-s − 6.51·49-s + 1.51·52-s + ⋯ |
L(s) = 1 | − 0.921·2-s − 0.151·4-s − 0.263·7-s + 1.06·8-s + 0.301·11-s − 1.38·13-s + 0.242·14-s − 0.825·16-s + 1.67·17-s − 0.229·19-s − 0.277·22-s − 1.52·23-s + 1.27·26-s + 0.0398·28-s − 0.168·29-s + 1.83·31-s − 0.300·32-s − 1.54·34-s − 0.393·37-s + 0.211·38-s + 0.875·41-s − 1.09·43-s − 0.0456·44-s + 1.40·46-s − 0.437·47-s − 0.930·49-s + 0.209·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 7 | \( 1 + 0.697T + 7T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 7.30T + 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.415399862659603199416534755798, −7.986651192879807112285336444773, −7.27442983818976872671281423815, −6.38967263571840803143135749724, −5.37353016843940857903175698802, −4.58610882863565380133372931648, −3.65481076245711625675108826063, −2.47159096994729384411981162893, −1.28350279549380974401560284608, 0,
1.28350279549380974401560284608, 2.47159096994729384411981162893, 3.65481076245711625675108826063, 4.58610882863565380133372931648, 5.37353016843940857903175698802, 6.38967263571840803143135749724, 7.27442983818976872671281423815, 7.986651192879807112285336444773, 8.415399862659603199416534755798