| L(s) = 1 | − 2·3-s + 4·5-s + 9-s − 8·15-s − 2·17-s − 2·19-s + 8·23-s + 11·25-s + 4·27-s − 2·29-s − 4·31-s − 6·37-s − 2·41-s − 8·43-s + 4·45-s + 4·47-s + 4·51-s − 10·53-s + 4·57-s − 6·59-s + 4·61-s − 12·67-s − 16·69-s − 14·73-s − 22·75-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s − 2.06·15-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 11/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.596·45-s + 0.583·47-s + 0.560·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s − 1.46·67-s − 1.92·69-s − 1.63·73-s − 2.54·75-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728588878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728588878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.52224774586160, −14.04011770394800, −13.36918012992139, −13.12137315990003, −12.59017386832308, −12.05190634792373, −11.30998475649300, −10.94920387374948, −10.45479341376518, −10.08699049406554, −9.347615994993423, −8.933790115880187, −8.529865007390213, −7.431950891771950, −6.880642249839994, −6.428972439125631, −5.952297343867077, −5.454374143255362, −4.945204547120235, −4.558716821553162, −3.352072317638810, −2.805158079523195, −1.893068582552557, −1.481546260111370, −0.4966677831453898,
0.4966677831453898, 1.481546260111370, 1.893068582552557, 2.805158079523195, 3.352072317638810, 4.558716821553162, 4.945204547120235, 5.454374143255362, 5.952297343867077, 6.428972439125631, 6.880642249839994, 7.431950891771950, 8.529865007390213, 8.933790115880187, 9.347615994993423, 10.08699049406554, 10.45479341376518, 10.94920387374948, 11.30998475649300, 12.05190634792373, 12.59017386832308, 13.12137315990003, 13.36918012992139, 14.04011770394800, 14.52224774586160
