| L(s) = 1 | + 2-s − 3-s − 6-s − 2·7-s + 8-s + 6·11-s + 13-s − 2·14-s − 16-s + 5·17-s − 2·19-s + 2·21-s + 6·22-s + 9·23-s − 24-s − 5·25-s + 26-s + 4·27-s − 2·29-s + 2·31-s − 6·32-s − 6·33-s + 5·34-s − 3·37-s − 2·38-s − 39-s + 2·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1.80·11-s + 0.277·13-s − 0.534·14-s − 1/4·16-s + 1.21·17-s − 0.458·19-s + 0.436·21-s + 1.27·22-s + 1.87·23-s − 0.204·24-s − 25-s + 0.196·26-s + 0.769·27-s − 0.371·29-s + 0.359·31-s − 1.06·32-s − 1.04·33-s + 0.857·34-s − 0.493·37-s − 0.324·38-s − 0.160·39-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283071607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283071607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 118 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−16.2543911815, −15.4764726943, −15.0549488724, −14.5672352322, −14.1141618451, −13.6104880262, −13.1844591499, −12.7212388089, −12.0398910945, −11.8505688387, −11.2223027943, −10.7139676344, −10.0483188287, −9.55616502519, −8.92164911100, −8.53451303300, −7.48066943653, −6.89480357690, −6.50787472122, −5.80610940498, −5.19054032448, −4.41225544554, −3.77806441304, −3.08142371802, −1.43353440781,
1.43353440781, 3.08142371802, 3.77806441304, 4.41225544554, 5.19054032448, 5.80610940498, 6.50787472122, 6.89480357690, 7.48066943653, 8.53451303300, 8.92164911100, 9.55616502519, 10.0483188287, 10.7139676344, 11.2223027943, 11.8505688387, 12.0398910945, 12.7212388089, 13.1844591499, 13.6104880262, 14.1141618451, 14.5672352322, 15.0549488724, 15.4764726943, 16.2543911815
