L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 6·13-s + 15-s + 4·17-s − 19-s + 2·21-s + 25-s − 27-s − 10·29-s + 2·31-s − 4·33-s + 2·35-s − 2·37-s − 6·39-s + 8·41-s + 8·43-s − 45-s − 3·49-s − 4·51-s − 6·53-s − 4·55-s + 57-s + 2·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.359·31-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.960·39-s + 1.24·41-s + 1.21·43-s − 0.149·45-s − 3/7·49-s − 0.560·51-s − 0.824·53-s − 0.539·55-s + 0.132·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.493707385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493707385\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.333036102517767066567699140688, −7.50211256028834193749536276691, −6.80148876070016204064607842568, −6.00644224995074401196591136573, −5.74317719875386558677938240888, −4.41437287658864631596915049052, −3.78118363667895262539592291166, −3.21174202037253558864752439073, −1.66505356482272892674763874367, −0.74263776044713023839304574067,
0.74263776044713023839304574067, 1.66505356482272892674763874367, 3.21174202037253558864752439073, 3.78118363667895262539592291166, 4.41437287658864631596915049052, 5.74317719875386558677938240888, 6.00644224995074401196591136573, 6.80148876070016204064607842568, 7.50211256028834193749536276691, 8.333036102517767066567699140688