| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 6·11-s + 4·13-s − 2·15-s − 17-s − 2·19-s − 21-s − 8·23-s − 25-s − 27-s + 8·31-s + 6·33-s + 2·35-s − 4·37-s − 4·39-s + 2·41-s + 12·43-s + 2·45-s + 8·47-s + 49-s + 51-s − 6·53-s − 12·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.43·31-s + 1.04·33-s + 0.338·35-s − 0.657·37-s − 0.640·39-s + 0.312·41-s + 1.82·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 1.61·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678645858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678645858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.093029078840248951332711412232, −7.51258306166415907211307723381, −6.43120424689753674457254897009, −5.92187087502871895273016517456, −5.42758354846672706331772857505, −4.60326137908628419092407155087, −3.79559352068166766121175455397, −2.51643039162554325427229286550, −1.97954728298424022151649414470, −0.69667989464450267401020853549,
0.69667989464450267401020853549, 1.97954728298424022151649414470, 2.51643039162554325427229286550, 3.79559352068166766121175455397, 4.60326137908628419092407155087, 5.42758354846672706331772857505, 5.92187087502871895273016517456, 6.43120424689753674457254897009, 7.51258306166415907211307723381, 8.093029078840248951332711412232
