L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·11-s + 6·13-s + 15-s − 2·17-s + 4·19-s + 21-s + 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s + 35-s + 6·37-s − 6·39-s + 6·41-s + 8·43-s − 45-s − 4·47-s + 49-s + 2·51-s − 2·53-s + 4·55-s − 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.960·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.106172139680155140027052621455, −7.53484534869047782039736472010, −6.75954644719482934446516169522, −5.79102887576166293832498984322, −5.46091756880018084862274164145, −4.28543211022430616838541340603, −3.61583481995535286108830734861, −2.62246999264770204206409429645, −1.26916896863925424658855324372, 0,
1.26916896863925424658855324372, 2.62246999264770204206409429645, 3.61583481995535286108830734861, 4.28543211022430616838541340603, 5.46091756880018084862274164145, 5.79102887576166293832498984322, 6.75954644719482934446516169522, 7.53484534869047782039736472010, 8.106172139680155140027052621455