L(s) = 1 | + 3-s + 4·5-s − 4·7-s + 9-s + 4·11-s − 4·13-s + 4·15-s + 6·17-s − 19-s − 4·21-s + 6·23-s + 11·25-s + 27-s + 2·29-s − 2·31-s + 4·33-s − 16·35-s + 4·37-s − 4·39-s − 6·41-s − 4·43-s + 4·45-s + 2·47-s + 9·49-s + 6·51-s − 6·53-s + 16·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 1.03·15-s + 1.45·17-s − 0.229·19-s − 0.872·21-s + 1.25·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.696·33-s − 2.70·35-s + 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.596·45-s + 0.291·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.393654167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.393654167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.865827742122993084197370629008, −9.414761335671312082745802899566, −8.838588343911370845805559800434, −7.33695141055614424124104099617, −6.58486416074202342376926146861, −5.93955197570232837725650602479, −4.90225789923401276368200956301, −3.39618467743782877483865031148, −2.67098425570496067707213534056, −1.38415691144283800743004390016,
1.38415691144283800743004390016, 2.67098425570496067707213534056, 3.39618467743782877483865031148, 4.90225789923401276368200956301, 5.93955197570232837725650602479, 6.58486416074202342376926146861, 7.33695141055614424124104099617, 8.838588343911370845805559800434, 9.414761335671312082745802899566, 9.865827742122993084197370629008