| L(s) = 1 | − 3-s − 7-s − 9-s + 4·11-s + 3·13-s − 7·17-s + 4·19-s + 21-s − 2·23-s − 8·29-s + 8·31-s − 4·33-s + 3·37-s − 3·39-s − 4·43-s − 5·47-s − 9·49-s + 7·51-s − 13·53-s − 4·57-s + 7·59-s − 10·61-s + 63-s − 19·67-s + 2·69-s + 10·71-s − 11·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s + 1.20·11-s + 0.832·13-s − 1.69·17-s + 0.917·19-s + 0.218·21-s − 0.417·23-s − 1.48·29-s + 1.43·31-s − 0.696·33-s + 0.493·37-s − 0.480·39-s − 0.609·43-s − 0.729·47-s − 9/7·49-s + 0.980·51-s − 1.78·53-s − 0.529·57-s + 0.911·59-s − 1.28·61-s + 0.125·63-s − 2.32·67-s + 0.240·69-s + 1.18·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 126 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 19 T + 220 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 172 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 178 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.52248461447981083271871007020, −7.16192065730929202213912097384, −6.63352652368889438771395198609, −6.47259830934224116917751372546, −6.18192597524370587940477169572, −6.11988834363186271989387523479, −5.46321738142674504754163132745, −5.22116037407795759947282835345, −4.72670475861424113589724985353, −4.33173971366049716673610485838, −4.19779893021655536674084439223, −3.58790204019342217368069486583, −3.22075698989748645280313087268, −3.06489936847121350526971444981, −2.29999803034415433291763479115, −1.93597908794572034982698454926, −1.37361243874609319956986379633, −1.07473623487783211915639187462, 0, 0,
1.07473623487783211915639187462, 1.37361243874609319956986379633, 1.93597908794572034982698454926, 2.29999803034415433291763479115, 3.06489936847121350526971444981, 3.22075698989748645280313087268, 3.58790204019342217368069486583, 4.19779893021655536674084439223, 4.33173971366049716673610485838, 4.72670475861424113589724985353, 5.22116037407795759947282835345, 5.46321738142674504754163132745, 6.11988834363186271989387523479, 6.18192597524370587940477169572, 6.47259830934224116917751372546, 6.63352652368889438771395198609, 7.16192065730929202213912097384, 7.52248461447981083271871007020
