| L(s) = 1 | − 3·2-s + 2·3-s + 8·4-s − 5·5-s − 6·6-s − 28·7-s − 45·8-s + 27·9-s + 15·10-s + 45·11-s + 16·12-s + 118·13-s + 84·14-s − 10·15-s + 135·16-s + 54·17-s − 81·18-s + 121·19-s − 40·20-s − 56·21-s − 135·22-s − 69·23-s − 90·24-s − 354·26-s + 154·27-s − 224·28-s − 324·29-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.384·3-s + 4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 1.98·8-s + 9-s + 0.474·10-s + 1.23·11-s + 0.384·12-s + 2.51·13-s + 1.60·14-s − 0.172·15-s + 2.10·16-s + 0.770·17-s − 1.06·18-s + 1.46·19-s − 0.447·20-s − 0.581·21-s − 1.30·22-s − 0.625·23-s − 0.765·24-s − 2.67·26-s + 1.09·27-s − 1.51·28-s − 2.07·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9285722971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9285722971\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T - 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 45 T + 694 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 54 T - 1997 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 121 T + 7782 T^{2} - 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 p T + 12 p^{2} T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 195 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 45 T - 101798 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 597 T + 207532 T^{2} + 597 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 360 T - 75779 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 392 T - 73317 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 280 T - 222363 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 668 T + 57207 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 782 T + 118485 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 768 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1194 T + 720667 T^{2} - 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
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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
−16.20348093144449504689622993338, −15.97486858880838963549489011226, −15.48839465610680338805068050048, −14.76672255783125311710237412717, −14.07891296923560549373955920596, −13.07312429038999064746194764810, −12.85526601304039970705123699336, −11.72722669157555856957103742189, −11.66213474839335432935209544351, −10.62658966317455537573195337626, −9.712394962297669992904599666843, −9.440871399908952146038413640350, −8.873277461279393451695739108945, −8.039056408159128828398033849895, −7.21481665755250255220144730026, −6.15988518812304639980879357328, −6.15486879486852257360359924076, −3.52376955375052533082259834212, −3.49957138537137634703060155228, −1.12683565678392103078838503175,
1.12683565678392103078838503175, 3.49957138537137634703060155228, 3.52376955375052533082259834212, 6.15486879486852257360359924076, 6.15988518812304639980879357328, 7.21481665755250255220144730026, 8.039056408159128828398033849895, 8.873277461279393451695739108945, 9.440871399908952146038413640350, 9.712394962297669992904599666843, 10.62658966317455537573195337626, 11.66213474839335432935209544351, 11.72722669157555856957103742189, 12.85526601304039970705123699336, 13.07312429038999064746194764810, 14.07891296923560549373955920596, 14.76672255783125311710237412717, 15.48839465610680338805068050048, 15.97486858880838963549489011226, 16.20348093144449504689622993338
