| L(s) = 1 | + 2·3-s − 4·5-s + 7-s + 9-s − 8·15-s + 2·17-s + 2·19-s + 2·21-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s − 4·35-s − 6·37-s − 2·41-s − 8·43-s − 4·45-s + 4·47-s + 49-s + 4·51-s − 10·53-s + 4·57-s + 6·59-s + 4·61-s + 63-s + 12·67-s − 14·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.312·41-s − 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s + 1.46·67-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118222343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118222343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.96173304669514, −12.34216682691483, −12.01034516724199, −11.46112054817778, −11.21352835087507, −10.69817722258404, −9.979737461980923, −9.626116917481915, −8.925836132135860, −8.488194699071257, −8.321946361368392, −7.785755413043857, −7.274613235210802, −7.133219410549301, −6.370205866226076, −5.449664231897214, −5.205779647128317, −4.402071483537586, −4.005860123988104, −3.476058802299579, −3.188185695567494, −2.608397601060162, −1.803872647753968, −1.232313454970452, −0.2741229970946460,
0.2741229970946460, 1.232313454970452, 1.803872647753968, 2.608397601060162, 3.188185695567494, 3.476058802299579, 4.005860123988104, 4.402071483537586, 5.205779647128317, 5.449664231897214, 6.370205866226076, 7.133219410549301, 7.274613235210802, 7.785755413043857, 8.321946361368392, 8.488194699071257, 8.925836132135860, 9.626116917481915, 9.979737461980923, 10.69817722258404, 11.21352835087507, 11.46112054817778, 12.01034516724199, 12.34216682691483, 12.96173304669514
