| L(s) = 1 | − 3-s + 9-s − 5·11-s − 13-s − 8·17-s − 5·19-s + 3·23-s − 27-s − 4·29-s + 6·31-s + 5·33-s + 3·37-s + 39-s + 7·41-s + 8·43-s + 47-s + 8·51-s + 9·53-s + 5·57-s + 12·59-s + 4·61-s + 8·67-s − 3·69-s + 2·71-s − 2·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 1.94·17-s − 1.14·19-s + 0.625·23-s − 0.192·27-s − 0.742·29-s + 1.07·31-s + 0.870·33-s + 0.493·37-s + 0.160·39-s + 1.09·41-s + 1.21·43-s + 0.145·47-s + 1.12·51-s + 1.23·53-s + 0.662·57-s + 1.56·59-s + 0.512·61-s + 0.977·67-s − 0.361·69-s + 0.237·71-s − 0.234·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.317839724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317839724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.96804308444092, −12.54748233652482, −11.99734164086730, −11.31974968664845, −11.08274748914122, −10.65121802938967, −10.28321108658473, −9.738251345286760, −9.072347047428269, −8.783423431074593, −8.146520871168093, −7.720299852266062, −7.162562553999583, −6.672899272327433, −6.279457574289643, −5.622813583317415, −5.250489130602621, −4.604606159551160, −4.307325887770489, −3.730630660629518, −2.769137894914055, −2.294348547020176, −2.114562320319201, −0.8583104298256530, −0.4100828175298242,
0.4100828175298242, 0.8583104298256530, 2.114562320319201, 2.294348547020176, 2.769137894914055, 3.730630660629518, 4.307325887770489, 4.604606159551160, 5.250489130602621, 5.622813583317415, 6.279457574289643, 6.672899272327433, 7.162562553999583, 7.720299852266062, 8.146520871168093, 8.783423431074593, 9.072347047428269, 9.738251345286760, 10.28321108658473, 10.65121802938967, 11.08274748914122, 11.31974968664845, 11.99734164086730, 12.54748233652482, 12.96804308444092
