| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 3·9-s − 3·11-s + 13-s − 4·15-s − 17-s + 5·19-s − 2·21-s − 23-s + 3·25-s + 4·27-s − 5·29-s + 7·31-s − 6·33-s + 2·35-s − 7·37-s + 2·39-s − 5·41-s − 16·43-s − 6·45-s + 47-s + 7·49-s − 2·51-s + 12·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s − 0.904·11-s + 0.277·13-s − 1.03·15-s − 0.242·17-s + 1.14·19-s − 0.436·21-s − 0.208·23-s + 3/5·25-s + 0.769·27-s − 0.928·29-s + 1.25·31-s − 1.04·33-s + 0.338·35-s − 1.15·37-s + 0.320·39-s − 0.780·41-s − 2.43·43-s − 0.894·45-s + 0.145·47-s + 49-s − 0.280·51-s + 1.64·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9539949981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9539949981\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_2$ | \( 1 + 16 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 7 T - 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.763948446136853868915146012502, −8.324667158009391933883234380201, −7.88015957608760105202312742878, −7.49361902728862253042698490210, −7.41515960356270389460315292879, −6.89851767334908217496606542213, −6.65614115853962734514606116445, −5.95135495486022931828543595627, −5.76781972019159061288266116323, −5.07377377997139603818937680317, −4.90183488400659615006908133477, −4.26834609183282095216384073222, −4.10309759386509739502640804763, −3.30731288219043116313833188952, −3.30639192611725997329959207048, −2.87427175041826401378324186435, −2.43487779921722442652741851143, −1.52084185058641043649189283630, −1.46506103660974575669545090414, −0.25221951214342277236937882900,
0.25221951214342277236937882900, 1.46506103660974575669545090414, 1.52084185058641043649189283630, 2.43487779921722442652741851143, 2.87427175041826401378324186435, 3.30639192611725997329959207048, 3.30731288219043116313833188952, 4.10309759386509739502640804763, 4.26834609183282095216384073222, 4.90183488400659615006908133477, 5.07377377997139603818937680317, 5.76781972019159061288266116323, 5.95135495486022931828543595627, 6.65614115853962734514606116445, 6.89851767334908217496606542213, 7.41515960356270389460315292879, 7.49361902728862253042698490210, 7.88015957608760105202312742878, 8.324667158009391933883234380201, 8.763948446136853868915146012502
