| L(s) = 1 | − 3·5-s − 5·11-s − 13-s − 17-s + 3·19-s − 5·23-s + 4·25-s + 2·29-s + 2·37-s + 7·41-s − 3·43-s − 12·47-s − 8·53-s + 15·55-s + 8·59-s − 6·61-s + 3·65-s + 8·67-s + 2·73-s − 12·79-s − 10·83-s + 3·85-s − 9·95-s − 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.50·11-s − 0.277·13-s − 0.242·17-s + 0.688·19-s − 1.04·23-s + 4/5·25-s + 0.371·29-s + 0.328·37-s + 1.09·41-s − 0.457·43-s − 1.75·47-s − 1.09·53-s + 2.02·55-s + 1.04·59-s − 0.768·61-s + 0.372·65-s + 0.977·67-s + 0.234·73-s − 1.35·79-s − 1.09·83-s + 0.325·85-s − 0.923·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1052188265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1052188265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.50361030099161, −12.95366122019002, −12.64037568579563, −12.10208214841095, −11.54941072172117, −11.27804269481024, −10.74649766990404, −10.12493375032201, −9.792405956064584, −9.177009373892842, −8.324465640374803, −8.142094396991050, −7.727218776054846, −7.270542272723035, −6.672289402487935, −6.033794917451022, −5.344185351938318, −4.958015293525451, −4.303555535899455, −3.893403436840747, −3.104863837675942, −2.771293152494480, −2.005473743234097, −1.111452306906678, −0.1068577563352411,
0.1068577563352411, 1.111452306906678, 2.005473743234097, 2.771293152494480, 3.104863837675942, 3.893403436840747, 4.303555535899455, 4.958015293525451, 5.344185351938318, 6.033794917451022, 6.672289402487935, 7.270542272723035, 7.727218776054846, 8.142094396991050, 8.324465640374803, 9.177009373892842, 9.792405956064584, 10.12493375032201, 10.74649766990404, 11.27804269481024, 11.54941072172117, 12.10208214841095, 12.64037568579563, 12.95366122019002, 13.50361030099161
