| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 4·13-s − 4·14-s + 5·16-s − 3·17-s − 7·19-s − 8·23-s + 8·26-s − 6·28-s − 2·29-s − 15·31-s + 6·32-s − 6·34-s − 4·37-s − 14·38-s + 9·41-s + 2·43-s − 16·46-s − 11·47-s − 11·49-s + 12·52-s + 5·53-s − 8·56-s − 4·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.727·17-s − 1.60·19-s − 1.66·23-s + 1.56·26-s − 1.13·28-s − 0.371·29-s − 2.69·31-s + 1.06·32-s − 1.02·34-s − 0.657·37-s − 2.27·38-s + 1.40·41-s + 0.304·43-s − 2.35·46-s − 1.60·47-s − 1.57·49-s + 1.66·52-s + 0.686·53-s − 1.06·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45562500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45562500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15 T + 107 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 123 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 111 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 218 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 115 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 189 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 139 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.52317529371126204947311102414, −7.45242511199219250958479778251, −6.80795006609542602217422968352, −6.78234766253139956663055279615, −6.18434345947349159476442909024, −6.05580473126149835490398655406, −5.62978232578177032356292723731, −5.59625747521485636438091253442, −4.73026169236372777252111834901, −4.58968271078764315868946480247, −4.06835807697890929010322525130, −3.90231973429489921628376001001, −3.36865601433828406348652877770, −3.31414536138633586769632675372, −2.38599403025724842211089213360, −2.36136849897668483134321974057, −1.55216453650957468102017351941, −1.53126458941738395741665518052, 0, 0,
1.53126458941738395741665518052, 1.55216453650957468102017351941, 2.36136849897668483134321974057, 2.38599403025724842211089213360, 3.31414536138633586769632675372, 3.36865601433828406348652877770, 3.90231973429489921628376001001, 4.06835807697890929010322525130, 4.58968271078764315868946480247, 4.73026169236372777252111834901, 5.59625747521485636438091253442, 5.62978232578177032356292723731, 6.05580473126149835490398655406, 6.18434345947349159476442909024, 6.78234766253139956663055279615, 6.80795006609542602217422968352, 7.45242511199219250958479778251, 7.52317529371126204947311102414
