| L(s) = 1 | − 3-s + 3·5-s − 9-s − 2·11-s + 4·13-s − 3·15-s + 6·17-s − 6·19-s + 3·23-s + 25-s + 6·29-s + 15·31-s + 2·33-s + 37-s − 4·39-s − 6·43-s − 3·45-s + 8·47-s − 14·49-s − 6·51-s − 6·55-s + 6·57-s + 13·59-s − 6·61-s + 12·65-s − 3·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.774·15-s + 1.45·17-s − 1.37·19-s + 0.625·23-s + 1/5·25-s + 1.11·29-s + 2.69·31-s + 0.348·33-s + 0.164·37-s − 0.640·39-s − 0.914·43-s − 0.447·45-s + 1.16·47-s − 2·49-s − 0.840·51-s − 0.809·55-s + 0.794·57-s + 1.69·59-s − 0.768·61-s + 1.48·65-s − 0.366·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673156847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673156847\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15 T + 114 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 194 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 176 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−11.71637930315705832383908896038, −11.32072987439750756990133324195, −10.55797848962274206480390322282, −10.31974112410654281292072961653, −10.06589824870708370937970806103, −9.600970605761722555739124403564, −8.720317118500883302013183754184, −8.639658734738239079781631052209, −8.017846064032105851810084781806, −7.52032751894912668633406699865, −6.51405201310665444503786570704, −6.42574959118232184729111613601, −5.85574000121122320487660419198, −5.57052142861751043992859836001, −4.80128707043152207413273055438, −4.39438615218903394977724440261, −3.30198380786633657028182233539, −2.81883750856868059259196740156, −1.93278793289298093103504980717, −1.00310502704205625408320475655,
1.00310502704205625408320475655, 1.93278793289298093103504980717, 2.81883750856868059259196740156, 3.30198380786633657028182233539, 4.39438615218903394977724440261, 4.80128707043152207413273055438, 5.57052142861751043992859836001, 5.85574000121122320487660419198, 6.42574959118232184729111613601, 6.51405201310665444503786570704, 7.52032751894912668633406699865, 8.017846064032105851810084781806, 8.639658734738239079781631052209, 8.720317118500883302013183754184, 9.600970605761722555739124403564, 10.06589824870708370937970806103, 10.31974112410654281292072961653, 10.55797848962274206480390322282, 11.32072987439750756990133324195, 11.71637930315705832383908896038
