| L(s) = 1 | − 2-s + 3·3-s + 2·4-s − 3·5-s − 3·6-s + 7-s − 5·8-s + 3·9-s + 3·10-s + 3·11-s + 6·12-s − 14-s − 9·15-s + 5·16-s + 2·17-s − 3·18-s + 19-s − 6·20-s + 3·21-s − 3·22-s − 15·24-s + 5·25-s + 2·28-s + 14·29-s + 9·30-s − 3·31-s − 10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 4-s − 1.34·5-s − 1.22·6-s + 0.377·7-s − 1.76·8-s + 9-s + 0.948·10-s + 0.904·11-s + 1.73·12-s − 0.267·14-s − 2.32·15-s + 5/4·16-s + 0.485·17-s − 0.707·18-s + 0.229·19-s − 1.34·20-s + 0.654·21-s − 0.639·22-s − 3.06·24-s + 25-s + 0.377·28-s + 2.59·29-s + 1.64·30-s − 0.538·31-s − 1.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.368076551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368076551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−10.06079988770602998195650327371, −9.589472085088987738886881322831, −8.880551276128510091210319388905, −8.601413937958611170742838679339, −8.457303233765370896824918830598, −7.960343463594599175013528876389, −7.924630696192998327259565704011, −7.13384198679466693510945879505, −6.81035681456118698446561915609, −6.51996515340616316203865059603, −6.01276094989407434318296635795, −5.06524301912545564168067974061, −4.92776849126758214528802585870, −3.90684296154350397270770949835, −3.50523013181163719205874532464, −3.44397887691165123366912573880, −2.53984078477029957018061134867, −2.51767491301095379565245519379, −1.50910939711207774995178201881, −0.68640472950616425433565807329,
0.68640472950616425433565807329, 1.50910939711207774995178201881, 2.51767491301095379565245519379, 2.53984078477029957018061134867, 3.44397887691165123366912573880, 3.50523013181163719205874532464, 3.90684296154350397270770949835, 4.92776849126758214528802585870, 5.06524301912545564168067974061, 6.01276094989407434318296635795, 6.51996515340616316203865059603, 6.81035681456118698446561915609, 7.13384198679466693510945879505, 7.924630696192998327259565704011, 7.960343463594599175013528876389, 8.457303233765370896824918830598, 8.601413937958611170742838679339, 8.880551276128510091210319388905, 9.589472085088987738886881322831, 10.06079988770602998195650327371
