| L(s) = 1 | − 2·4-s + 3·5-s + 7-s + 11-s + 4·13-s + 4·16-s − 6·20-s + 6·23-s + 4·25-s − 2·28-s − 6·29-s − 5·31-s + 3·35-s − 8·37-s + 9·41-s + 2·43-s − 2·44-s + 9·47-s + 49-s − 8·52-s + 3·53-s + 3·55-s − 7·61-s − 8·64-s + 12·65-s + 4·67-s + 9·71-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s + 0.377·7-s + 0.301·11-s + 1.10·13-s + 16-s − 1.34·20-s + 1.25·23-s + 4/5·25-s − 0.377·28-s − 1.11·29-s − 0.898·31-s + 0.507·35-s − 1.31·37-s + 1.40·41-s + 0.304·43-s − 0.301·44-s + 1.31·47-s + 1/7·49-s − 1.10·52-s + 0.412·53-s + 0.404·55-s − 0.896·61-s − 64-s + 1.48·65-s + 0.488·67-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.579494200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.579494200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.07033768011157, −12.50440951798640, −12.11481289625522, −11.25386491911961, −10.93719806498118, −10.57596546406293, −10.00455878449518, −9.445772594857106, −9.093493129257189, −8.887017415948365, −8.362610241440936, −7.675263458933780, −7.210116411865029, −6.628931902514009, −5.967895631431285, −5.590229637146446, −5.364775323166169, −4.676047112816173, −4.094476064607505, −3.637198960093936, −3.053963080865493, −2.281521845371744, −1.679771531452553, −1.192025180283198, −0.5633105501998239,
0.5633105501998239, 1.192025180283198, 1.679771531452553, 2.281521845371744, 3.053963080865493, 3.637198960093936, 4.094476064607505, 4.676047112816173, 5.364775323166169, 5.590229637146446, 5.967895631431285, 6.628931902514009, 7.210116411865029, 7.675263458933780, 8.362610241440936, 8.887017415948365, 9.093493129257189, 9.445772594857106, 10.00455878449518, 10.57596546406293, 10.93719806498118, 11.25386491911961, 12.11481289625522, 12.50440951798640, 13.07033768011157
