| L(s) = 1 | − 222.·7-s − 682.·11-s + 412.·13-s + 866.·17-s + 668.·19-s − 3.32e3·23-s − 1.25e3·29-s − 9.70e3·31-s − 8.80e3·37-s − 6.73e3·41-s − 1.70e4·43-s − 1.29e4·47-s + 3.25e4·49-s − 4.15e3·53-s + 2.84e4·59-s + 1.03e4·61-s − 6.59e4·67-s + 2.00e4·71-s + 4.71e4·73-s + 1.51e5·77-s − 2.10e4·79-s + 9.03e4·83-s + 4.15e4·89-s − 9.15e4·91-s − 5.63e4·97-s + 1.44e4·101-s − 8.58e4·103-s + ⋯ |
| L(s) = 1 | − 1.71·7-s − 1.70·11-s + 0.676·13-s + 0.726·17-s + 0.424·19-s − 1.31·23-s − 0.276·29-s − 1.81·31-s − 1.05·37-s − 0.625·41-s − 1.40·43-s − 0.856·47-s + 1.93·49-s − 0.203·53-s + 1.06·59-s + 0.355·61-s − 1.79·67-s + 0.471·71-s + 1.03·73-s + 2.91·77-s − 0.379·79-s + 1.43·83-s + 0.556·89-s − 1.15·91-s − 0.608·97-s + 0.141·101-s − 0.797·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5469158452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5469158452\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 222.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 682.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 412.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 866.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 668.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.15e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.63e4T + 8.58e9T^{2} \) |
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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
−9.562214709014832578862020475757, −8.508370662190982814849215427315, −7.66401754411429145336441258004, −6.79462072773645648758369828161, −5.85316333006103464303381046067, −5.21940499307802787628974299636, −3.64823917566357313861137585330, −3.16692123737246056024578809537, −1.93917526723945490780752180278, −0.30984550727399831310325863039,
0.30984550727399831310325863039, 1.93917526723945490780752180278, 3.16692123737246056024578809537, 3.64823917566357313861137585330, 5.21940499307802787628974299636, 5.85316333006103464303381046067, 6.79462072773645648758369828161, 7.66401754411429145336441258004, 8.508370662190982814849215427315, 9.562214709014832578862020475757
