| L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 13-s − 17-s + 4·19-s + 21-s − 3·23-s − 27-s + 8·29-s − 4·31-s − 33-s − 3·37-s − 39-s − 9·41-s + 8·43-s + 10·47-s − 6·49-s + 51-s + 53-s − 4·57-s − 4·59-s + 11·61-s − 63-s + 4·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.174·33-s − 0.493·37-s − 0.160·39-s − 1.40·41-s + 1.21·43-s + 1.45·47-s − 6/7·49-s + 0.140·51-s + 0.137·53-s − 0.529·57-s − 0.520·59-s + 1.40·61-s − 0.125·63-s + 0.488·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−14.27999198177489, −14.06071109555084, −13.57019717684944, −12.93112750611078, −12.39055072696243, −12.04498299462212, −11.55702090359412, −10.90780888886687, −10.58006321674067, −9.843306914241610, −9.583712378479876, −8.897355800276586, −8.313575606336562, −7.817804063791085, −6.967363087216756, −6.778003578983201, −6.139044565921116, −5.429153154436272, −5.185432208760575, −4.215820034555625, −3.897461927357273, −3.097878970704709, −2.458415772550647, −1.559099603216855, −0.9076848322465135, 0,
0.9076848322465135, 1.559099603216855, 2.458415772550647, 3.097878970704709, 3.897461927357273, 4.215820034555625, 5.185432208760575, 5.429153154436272, 6.139044565921116, 6.778003578983201, 6.967363087216756, 7.817804063791085, 8.313575606336562, 8.897355800276586, 9.583712378479876, 9.843306914241610, 10.58006321674067, 10.90780888886687, 11.55702090359412, 12.04498299462212, 12.39055072696243, 12.93112750611078, 13.57019717684944, 14.06071109555084, 14.27999198177489
