| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s + 4·8-s + 4·10-s − 6·12-s + 4·13-s + 4·14-s − 4·15-s + 5·16-s + 2·17-s − 4·19-s + 6·20-s − 4·21-s + 2·23-s − 8·24-s − 4·25-s + 8·26-s + 2·27-s + 6·28-s + 16·29-s − 8·30-s − 10·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 1.26·10-s − 1.73·12-s + 1.10·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.917·19-s + 1.34·20-s − 0.872·21-s + 0.417·23-s − 1.63·24-s − 4/5·25-s + 1.56·26-s + 0.384·27-s + 1.13·28-s + 2.97·29-s − 1.46·30-s − 1.79·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.238227083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.238227083\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 232 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 240 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−11.70067506929354483704720725075, −11.58633516157768698972319476615, −11.11381917331582317230341312977, −10.67695257272027306870633664533, −10.23218977759189274663605114359, −9.910464446373822919909083333012, −8.773354454877926775250115319318, −8.755371363738823579525397429705, −7.84429094861647662196212062621, −7.44881522608849169141167226406, −6.51333532832681734755701038830, −6.31971864967619896396942253525, −5.82809701286436456124192797961, −5.58202829084117856042817900705, −4.74557933607194402922955567371, −4.62123766791621609842417052097, −3.70438461595417975502626028657, −2.99236055587300241757013965243, −2.11107970094479299698612823224, −1.30103763604643903910844266711,
1.30103763604643903910844266711, 2.11107970094479299698612823224, 2.99236055587300241757013965243, 3.70438461595417975502626028657, 4.62123766791621609842417052097, 4.74557933607194402922955567371, 5.58202829084117856042817900705, 5.82809701286436456124192797961, 6.31971864967619896396942253525, 6.51333532832681734755701038830, 7.44881522608849169141167226406, 7.84429094861647662196212062621, 8.755371363738823579525397429705, 8.773354454877926775250115319318, 9.910464446373822919909083333012, 10.23218977759189274663605114359, 10.67695257272027306870633664533, 11.11381917331582317230341312977, 11.58633516157768698972319476615, 11.70067506929354483704720725075
