| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·13-s + 15-s + 6·17-s + 2·19-s + 2·21-s − 4·23-s + 25-s − 27-s + 2·29-s + 8·31-s + 2·35-s + 10·37-s − 4·39-s − 2·41-s − 6·43-s − 45-s + 8·47-s − 3·49-s − 6·51-s − 10·53-s − 2·57-s + 12·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s + 0.258·15-s + 1.45·17-s + 0.458·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.338·35-s + 1.64·37-s − 0.640·39-s − 0.312·41-s − 0.914·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.840·51-s − 1.37·53-s − 0.264·57-s + 1.53·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853123896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853123896\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−15.54013499794766, −14.56276550373344, −14.13928371707947, −13.52460847456789, −13.00670405956707, −12.41056607345738, −12.02595813467088, −11.40468294838157, −11.06048791940283, −10.12054561417965, −9.968352233381487, −9.385397980740270, −8.429360767816772, −8.107630261317427, −7.479487134032959, −6.743285218882828, −6.155739706516281, −5.861698365501734, −5.012519516150897, −4.393018454337787, −3.539234871295239, −3.298382245945970, −2.289407646490224, −1.152373806767827, −0.6482356099786255,
0.6482356099786255, 1.152373806767827, 2.289407646490224, 3.298382245945970, 3.539234871295239, 4.393018454337787, 5.012519516150897, 5.861698365501734, 6.155739706516281, 6.743285218882828, 7.479487134032959, 8.107630261317427, 8.429360767816772, 9.385397980740270, 9.968352233381487, 10.12054561417965, 11.06048791940283, 11.40468294838157, 12.02595813467088, 12.41056607345738, 13.00670405956707, 13.52460847456789, 14.13928371707947, 14.56276550373344, 15.54013499794766
