| L(s) = 1 | + 4-s − 2·7-s + 9-s − 8·11-s + 12·13-s + 16-s + 4·17-s − 6·25-s − 2·28-s − 4·29-s + 36-s − 20·37-s − 8·43-s − 8·44-s + 3·49-s + 12·52-s + 6·53-s + 8·59-s − 2·63-s + 64-s + 4·68-s + 16·77-s + 81-s − 12·89-s − 24·91-s − 28·97-s − 8·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s + 1/3·9-s − 2.41·11-s + 3.32·13-s + 1/4·16-s + 0.970·17-s − 6/5·25-s − 0.377·28-s − 0.742·29-s + 1/6·36-s − 3.28·37-s − 1.21·43-s − 1.20·44-s + 3/7·49-s + 1.66·52-s + 0.824·53-s + 1.04·59-s − 0.251·63-s + 1/8·64-s + 0.485·68-s + 1.82·77-s + 1/9·81-s − 1.27·89-s − 2.51·91-s − 2.84·97-s − 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4955076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4955076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659141855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659141855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.31618436492604863435902543640, −6.84552480602986516155667793601, −6.56225054608614269754758337655, −6.05857616111761188736923156685, −5.61588131716717336152713515261, −5.33985014787602837985094112170, −5.24124351541382054400781904279, −4.16551102118751694100907626234, −3.67268213755857288293135595950, −3.62482887081886485478101246558, −3.02877457374453947654735474964, −2.58459732524656909069587264431, −1.64795775077704015094055119225, −1.55929188669763310789284398036, −0.43629256145672715618319237912,
0.43629256145672715618319237912, 1.55929188669763310789284398036, 1.64795775077704015094055119225, 2.58459732524656909069587264431, 3.02877457374453947654735474964, 3.62482887081886485478101246558, 3.67268213755857288293135595950, 4.16551102118751694100907626234, 5.24124351541382054400781904279, 5.33985014787602837985094112170, 5.61588131716717336152713515261, 6.05857616111761188736923156685, 6.56225054608614269754758337655, 6.84552480602986516155667793601, 7.31618436492604863435902543640
