| L(s) = 1 | + 2·2-s − 3·3-s − 4-s + 3·5-s − 6·6-s − 8·8-s + 3·9-s + 6·10-s + 3·11-s + 3·12-s + 2·13-s − 9·15-s − 7·16-s + 4·17-s + 6·18-s − 19-s − 3·20-s + 6·22-s + 24·24-s + 5·25-s + 4·26-s − 7·29-s − 18·30-s + 3·31-s + 14·32-s − 9·33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s − 1/2·4-s + 1.34·5-s − 2.44·6-s − 2.82·8-s + 9-s + 1.89·10-s + 0.904·11-s + 0.866·12-s + 0.554·13-s − 2.32·15-s − 7/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s − 0.670·20-s + 1.27·22-s + 4.89·24-s + 25-s + 0.784·26-s − 1.29·29-s − 3.28·30-s + 0.538·31-s + 2.47·32-s − 1.56·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510607936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510607936\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.97362943160390685804829381039, −10.40862877019752126746322511114, −9.922556701233750088242751248758, −9.600246175692059040102002772527, −9.069295824135164454064268136349, −8.949953815024556367006728802541, −8.253635428651120014387623823920, −7.59423588506133414293067071337, −6.83558622988435071504855307682, −6.19608207106081283978248930988, −6.11055666883225270480791372074, −5.64632314701751292668146259107, −5.52199942051984384581974102511, −4.99440595445804030262982022108, −4.24172625753363698227480771009, −4.17980600137660020231207138725, −3.28969065644276697809865200440, −2.75810618453237808642489293905, −1.55535963690720593600409619946, −0.63879991993342906314309035535,
0.63879991993342906314309035535, 1.55535963690720593600409619946, 2.75810618453237808642489293905, 3.28969065644276697809865200440, 4.17980600137660020231207138725, 4.24172625753363698227480771009, 4.99440595445804030262982022108, 5.52199942051984384581974102511, 5.64632314701751292668146259107, 6.11055666883225270480791372074, 6.19608207106081283978248930988, 6.83558622988435071504855307682, 7.59423588506133414293067071337, 8.253635428651120014387623823920, 8.949953815024556367006728802541, 9.069295824135164454064268136349, 9.600246175692059040102002772527, 9.922556701233750088242751248758, 10.40862877019752126746322511114, 10.97362943160390685804829381039
