| L(s) = 1 | − 2-s − 2·5-s − 8-s + 9-s + 2·10-s − 6·11-s + 6·13-s − 16-s + 3·17-s − 18-s − 2·19-s + 6·22-s + 2·25-s − 6·26-s + 10·29-s − 4·31-s + 6·32-s − 3·34-s − 3·37-s + 2·38-s + 2·40-s − 7·41-s − 2·43-s − 2·45-s + 12·47-s − 4·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s + 1.66·13-s − 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s + 1.27·22-s + 2/5·25-s − 1.17·26-s + 1.85·29-s − 0.718·31-s + 1.06·32-s − 0.514·34-s − 0.493·37-s + 0.324·38-s + 0.316·40-s − 1.09·41-s − 0.304·43-s − 0.298·45-s + 1.75·47-s − 4/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 745 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 745 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3033683921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3033683921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 149 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 58 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 106 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.0175590619, −18.7852639927, −18.3864789002, −17.8724895485, −17.2471813924, −16.1863564617, −16.0022669588, −15.4897891357, −14.9633152762, −13.8010550729, −13.4674608833, −12.5770536509, −12.0604944284, −11.1426095747, −10.5931757928, −10.0239343547, −8.90231813249, −8.36451151583, −7.84075246067, −6.82992066661, −5.75608613814, −4.54479150477, −3.18405320212,
3.18405320212, 4.54479150477, 5.75608613814, 6.82992066661, 7.84075246067, 8.36451151583, 8.90231813249, 10.0239343547, 10.5931757928, 11.1426095747, 12.0604944284, 12.5770536509, 13.4674608833, 13.8010550729, 14.9633152762, 15.4897891357, 16.0022669588, 16.1863564617, 17.2471813924, 17.8724895485, 18.3864789002, 18.7852639927, 19.0175590619
