| L(s) = 1 | − 2·4-s + 11-s − 2·13-s + 4·16-s − 8·17-s + 4·19-s − 2·23-s − 3·31-s − 3·37-s + 2·41-s + 5·43-s − 2·44-s + 6·47-s + 4·52-s + 4·53-s − 10·59-s − 7·61-s − 8·64-s + 4·67-s + 16·68-s − 6·71-s + 73-s − 8·76-s − 5·79-s + 2·83-s + 8·89-s + 4·92-s + ⋯ |
| L(s) = 1 | − 4-s + 0.301·11-s − 0.554·13-s + 16-s − 1.94·17-s + 0.917·19-s − 0.417·23-s − 0.538·31-s − 0.493·37-s + 0.312·41-s + 0.762·43-s − 0.301·44-s + 0.875·47-s + 0.554·52-s + 0.549·53-s − 1.30·59-s − 0.896·61-s − 64-s + 0.488·67-s + 1.94·68-s − 0.712·71-s + 0.117·73-s − 0.917·76-s − 0.562·79-s + 0.219·83-s + 0.847·89-s + 0.417·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.84172921153016, −13.43664275417756, −12.83057594074549, −12.41043837722118, −11.98448496013362, −11.36338903659514, −10.85948598419583, −10.39246481851271, −9.798516961529297, −9.284929545852598, −8.980235775896834, −8.590476905987749, −7.831976928400632, −7.442023605758631, −6.889800647659821, −6.238184436002854, −5.696858129713354, −5.169284346349731, −4.497525237701395, −4.280380322236443, −3.595170257535861, −2.941376693646828, −2.240972586175608, −1.568483091308237, −0.6885230933753313, 0,
0.6885230933753313, 1.568483091308237, 2.240972586175608, 2.941376693646828, 3.595170257535861, 4.280380322236443, 4.497525237701395, 5.169284346349731, 5.696858129713354, 6.238184436002854, 6.889800647659821, 7.442023605758631, 7.831976928400632, 8.590476905987749, 8.980235775896834, 9.284929545852598, 9.798516961529297, 10.39246481851271, 10.85948598419583, 11.36338903659514, 11.98448496013362, 12.41043837722118, 12.83057594074549, 13.43664275417756, 13.84172921153016
