| L(s) = 1 | − 3-s − 4·5-s + 4·7-s + 9-s − 13-s + 4·15-s + 2·17-s − 2·19-s − 4·21-s − 6·23-s + 11·25-s − 27-s − 2·29-s + 8·31-s − 16·35-s − 2·37-s + 39-s + 10·41-s + 2·43-s − 4·45-s − 2·47-s + 9·49-s − 2·51-s − 4·53-s + 2·57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s − 0.872·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 2.70·35-s − 0.328·37-s + 0.160·39-s + 1.56·41-s + 0.304·43-s − 0.596·45-s − 0.291·47-s + 9/7·49-s − 0.280·51-s − 0.549·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359099528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359099528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.45289981373020, −11.98021758100298, −11.88603060073869, −11.28834068125490, −11.05838691874921, −10.61957915295467, −10.02258144130134, −9.580631437213228, −8.667108506160268, −8.440659275660310, −7.998318795847383, −7.599424337204353, −7.300103912157768, −6.670683091946945, −6.057304472617094, −5.502091367727032, −4.950990111257364, −4.488302292699874, −4.146008823725870, −3.777003157825609, −2.960685753673953, −2.351336036539822, −1.617383375786112, −0.9659145085610193, −0.3914530601059633,
0.3914530601059633, 0.9659145085610193, 1.617383375786112, 2.351336036539822, 2.960685753673953, 3.777003157825609, 4.146008823725870, 4.488302292699874, 4.950990111257364, 5.502091367727032, 6.057304472617094, 6.670683091946945, 7.300103912157768, 7.599424337204353, 7.998318795847383, 8.440659275660310, 8.667108506160268, 9.580631437213228, 10.02258144130134, 10.61957915295467, 11.05838691874921, 11.28834068125490, 11.88603060073869, 11.98021758100298, 12.45289981373020
