| L(s) = 1 | + 3-s + 2·7-s + 9-s − 6·11-s + 2·13-s + 2·17-s − 2·19-s + 2·21-s − 6·23-s + 27-s + 10·29-s − 4·31-s − 6·33-s − 2·37-s + 2·39-s + 6·41-s − 6·47-s − 3·49-s + 2·51-s + 10·53-s − 2·57-s + 6·59-s + 6·61-s + 2·63-s + 16·67-s − 6·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.436·21-s − 1.25·23-s + 0.192·27-s + 1.85·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s − 0.264·57-s + 0.781·59-s + 0.768·61-s + 0.251·63-s + 1.95·67-s − 0.722·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.528447587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.528447587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.923661958602703267832077222595, −7.17976863666379635691209066040, −6.32824585757925479626570100293, −5.50937528530483389902804866713, −4.96734436820679151351537720624, −4.18409610846266630219553195159, −3.38346014800829401115930133420, −2.50173208265971993450259785203, −1.94556142796995841440391766954, −0.72563287055985241186247983205,
0.72563287055985241186247983205, 1.94556142796995841440391766954, 2.50173208265971993450259785203, 3.38346014800829401115930133420, 4.18409610846266630219553195159, 4.96734436820679151351537720624, 5.50937528530483389902804866713, 6.32824585757925479626570100293, 7.17976863666379635691209066040, 7.923661958602703267832077222595
