| L(s) = 1 | + 2-s + 3·3-s − 2·4-s + 2·5-s + 3·6-s + 7-s − 3·8-s + 2·9-s + 2·10-s − 6·11-s − 6·12-s + 8·13-s + 14-s + 6·15-s + 16-s + 11·17-s + 2·18-s − 5·19-s − 4·20-s + 3·21-s − 6·22-s + 8·23-s − 9·24-s − 2·25-s + 8·26-s − 6·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 4-s + 0.894·5-s + 1.22·6-s + 0.377·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s − 1.80·11-s − 1.73·12-s + 2.21·13-s + 0.267·14-s + 1.54·15-s + 1/4·16-s + 2.66·17-s + 0.471·18-s − 1.14·19-s − 0.894·20-s + 0.654·21-s − 1.27·22-s + 1.66·23-s − 1.83·24-s − 2/5·25-s + 1.56·26-s − 1.15·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865156371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865156371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 211 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 147 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 127 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−12.94688230149341132220237572836, −12.60765245875506820186987410092, −11.73793555735758583432511131487, −11.00946118260807241438039653299, −10.40572907488464236404775667184, −10.35441309180127155496159833846, −9.250352518181498519287741720120, −9.179824028246581985165728588630, −8.625227336884577484639808350636, −8.366610100307217069606563423091, −7.64720430681378228300193610192, −7.43849445532593941380600467855, −5.95565940507301218652034508117, −5.73413738679359382492653842031, −5.35409419253810527228981723930, −4.53165472346215825844321097401, −3.53059258412759958144996583971, −3.41443173535724879756555829508, −2.65431700043165294476165593416, −1.58038285544726488074611921013,
1.58038285544726488074611921013, 2.65431700043165294476165593416, 3.41443173535724879756555829508, 3.53059258412759958144996583971, 4.53165472346215825844321097401, 5.35409419253810527228981723930, 5.73413738679359382492653842031, 5.95565940507301218652034508117, 7.43849445532593941380600467855, 7.64720430681378228300193610192, 8.366610100307217069606563423091, 8.625227336884577484639808350636, 9.179824028246581985165728588630, 9.250352518181498519287741720120, 10.35441309180127155496159833846, 10.40572907488464236404775667184, 11.00946118260807241438039653299, 11.73793555735758583432511131487, 12.60765245875506820186987410092, 12.94688230149341132220237572836
