| L(s) = 1 | + 3-s + 2·5-s − 5·7-s − 9-s − 2·11-s − 6·13-s + 2·15-s − 3·17-s + 7·19-s − 5·21-s + 6·23-s + 3·25-s − 13·29-s + 7·31-s − 2·33-s − 10·35-s − 19·37-s − 6·39-s − 8·41-s + 2·43-s − 2·45-s − 2·47-s + 9·49-s − 3·51-s − 7·53-s − 4·55-s + 7·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.88·7-s − 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.516·15-s − 0.727·17-s + 1.60·19-s − 1.09·21-s + 1.25·23-s + 3/5·25-s − 2.41·29-s + 1.25·31-s − 0.348·33-s − 1.69·35-s − 3.12·37-s − 0.960·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s − 0.291·47-s + 9/7·49-s − 0.420·51-s − 0.961·53-s − 0.539·55-s + 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19 T + 160 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 80 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 21 T + 228 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.499265568129977361205598032197, −8.081849035586042927734999516818, −7.42211469378310476126837818669, −7.28140531536497405301610553488, −6.80659910142652244625359351937, −6.79005615202367073744992693170, −6.17193451594584493311407235767, −5.71789778929994399575546332361, −5.21806970688879035756218153459, −5.15110761257051236473884189518, −4.79290748157005225642075227038, −3.90086229137595267346150829691, −3.43192700173976050549349821681, −3.26261732832436156828400407374, −2.65034276651011634087465381970, −2.57494878321239347086960177201, −1.89286048047060019269981260481, −1.26279422969758656554803887559, 0, 0,
1.26279422969758656554803887559, 1.89286048047060019269981260481, 2.57494878321239347086960177201, 2.65034276651011634087465381970, 3.26261732832436156828400407374, 3.43192700173976050549349821681, 3.90086229137595267346150829691, 4.79290748157005225642075227038, 5.15110761257051236473884189518, 5.21806970688879035756218153459, 5.71789778929994399575546332361, 6.17193451594584493311407235767, 6.79005615202367073744992693170, 6.80659910142652244625359351937, 7.28140531536497405301610553488, 7.42211469378310476126837818669, 8.081849035586042927734999516818, 8.499265568129977361205598032197
