| L(s) = 1 | + 3·4-s − 6·7-s + 4·13-s + 5·16-s + 2·19-s − 18·28-s − 6·31-s + 2·37-s − 2·43-s + 13·49-s + 12·52-s + 14·61-s + 3·64-s − 24·67-s + 22·73-s + 6·76-s + 30·79-s − 24·91-s + 14·97-s + 14·103-s + 34·109-s − 30·112-s + 6·121-s − 18·124-s + 127-s + 131-s − 12·133-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 2.26·7-s + 1.10·13-s + 5/4·16-s + 0.458·19-s − 3.40·28-s − 1.07·31-s + 0.328·37-s − 0.304·43-s + 13/7·49-s + 1.66·52-s + 1.79·61-s + 3/8·64-s − 2.93·67-s + 2.57·73-s + 0.688·76-s + 3.37·79-s − 2.51·91-s + 1.42·97-s + 1.37·103-s + 3.25·109-s − 2.83·112-s + 6/11·121-s − 1.61·124-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.057706977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057706977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.01949562177065143244385887088, −10.26970462124441613773932271582, −9.753131323959798992367574928052, −9.716433023315420714720090309443, −8.854970762158158320656806156993, −8.832137915590806122752744179632, −7.971496139426878862089246138259, −7.35658528880860920168453168465, −7.30537812173058972741068848350, −6.45025569392483847593973729007, −6.35866305503470250365889135222, −6.16482571886967247338423844759, −5.48578857875682619499962455932, −4.86773569835033415573584131310, −3.85601571529966850519457451903, −3.35304967251654141077534059260, −3.30375076762567740142856987476, −2.44210086430761096782746783788, −1.87220090675730271137255508551, −0.74637271727844988252325148508,
0.74637271727844988252325148508, 1.87220090675730271137255508551, 2.44210086430761096782746783788, 3.30375076762567740142856987476, 3.35304967251654141077534059260, 3.85601571529966850519457451903, 4.86773569835033415573584131310, 5.48578857875682619499962455932, 6.16482571886967247338423844759, 6.35866305503470250365889135222, 6.45025569392483847593973729007, 7.30537812173058972741068848350, 7.35658528880860920168453168465, 7.971496139426878862089246138259, 8.832137915590806122752744179632, 8.854970762158158320656806156993, 9.716433023315420714720090309443, 9.753131323959798992367574928052, 10.26970462124441613773932271582, 11.01949562177065143244385887088
