L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 6·13-s + 2·17-s + 8·19-s + 21-s + 4·23-s − 27-s − 2·29-s − 8·31-s + 33-s + 6·37-s + 6·39-s − 2·41-s + 8·43-s + 4·47-s + 49-s − 2·51-s + 2·53-s − 8·57-s + 12·59-s − 10·61-s − 63-s − 12·67-s − 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.960·39-s − 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.125·63-s − 1.46·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.72025609598187, −12.14384734501433, −11.85667014997716, −11.53959437340626, −10.88594872526983, −10.40823584696006, −10.09854358057334, −9.545699663027192, −9.155738612739883, −8.922767457796704, −7.820553450471296, −7.605055361334141, −7.255357177748037, −6.928852852829962, −6.067872040190286, −5.694556030855364, −5.303252127071613, −4.859746253714115, −4.342938034773343, −3.686763536402856, −3.047776493394561, −2.732343541804764, −2.026946320764803, −1.282160957065272, −0.6847753455875377, 0,
0.6847753455875377, 1.282160957065272, 2.026946320764803, 2.732343541804764, 3.047776493394561, 3.686763536402856, 4.342938034773343, 4.859746253714115, 5.303252127071613, 5.694556030855364, 6.067872040190286, 6.928852852829962, 7.255357177748037, 7.605055361334141, 7.820553450471296, 8.922767457796704, 9.155738612739883, 9.545699663027192, 10.09854358057334, 10.40823584696006, 10.88594872526983, 11.53959437340626, 11.85667014997716, 12.14384734501433, 12.72025609598187