| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 5-s + 4·6-s − 4·8-s + 3·9-s − 2·10-s − 6·12-s − 5·13-s − 2·15-s + 5·16-s − 6·18-s − 2·19-s + 3·20-s − 2·23-s + 8·24-s + 25-s + 10·26-s − 4·27-s + 7·29-s + 4·30-s − 12·31-s − 6·32-s + 9·36-s + 37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 1.41·8-s + 9-s − 0.632·10-s − 1.73·12-s − 1.38·13-s − 0.516·15-s + 5/4·16-s − 1.41·18-s − 0.458·19-s + 0.670·20-s − 0.417·23-s + 1.63·24-s + 1/5·25-s + 1.96·26-s − 0.769·27-s + 1.29·29-s + 0.730·30-s − 2.15·31-s − 1.06·32-s + 3/2·36-s + 0.164·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6033429073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6033429073\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - T + 64 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 82 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 174 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.104604702586528655906445633738, −7.83134093719014011538209227947, −7.29569932191362623130044007211, −7.25186462971707864819415288070, −6.71499611324612220429945360873, −6.61567877628310367378312998863, −6.07756409692961215654746137915, −5.85695241198848078830704039583, −5.32136237583507011365272812964, −5.14341383306162445390040615121, −4.73865508183305681634121369341, −4.26183198850873704387624631815, −3.72988250992142800093473741792, −3.34252058310571327078580677716, −2.63960247895371269782993674535, −2.37242954541483000747713821482, −1.80446741879818055301256070890, −1.57682909990166113307177242521, −0.71510177643000334836138940405, −0.37753831988603133028920515211,
0.37753831988603133028920515211, 0.71510177643000334836138940405, 1.57682909990166113307177242521, 1.80446741879818055301256070890, 2.37242954541483000747713821482, 2.63960247895371269782993674535, 3.34252058310571327078580677716, 3.72988250992142800093473741792, 4.26183198850873704387624631815, 4.73865508183305681634121369341, 5.14341383306162445390040615121, 5.32136237583507011365272812964, 5.85695241198848078830704039583, 6.07756409692961215654746137915, 6.61567877628310367378312998863, 6.71499611324612220429945360873, 7.25186462971707864819415288070, 7.29569932191362623130044007211, 7.83134093719014011538209227947, 8.104604702586528655906445633738
