| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s − 4·11-s + 12-s + 14-s + 15-s − 16-s + 17-s − 18-s + 20-s + 21-s + 4·22-s − 4·23-s − 3·24-s + 25-s − 27-s + 28-s − 2·29-s − 30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.12746433424192, −12.71922845826293, −12.17905226505455, −11.77791327944854, −11.13275422995836, −10.79395602331628, −10.34719019491919, −9.892737197854014, −9.656754197780199, −8.960203935747749, −8.439979705290436, −8.191839130706668, −7.562886353420246, −7.199697577587415, −6.813780848045421, −6.008824186487559, −5.412231847248797, −5.288265027615739, −4.586945406780664, −4.032632801501248, −3.593896350074533, −2.987319184037793, −2.144629183943468, −1.654613916601491, −0.8498641617644105, 0, 0,
0.8498641617644105, 1.654613916601491, 2.144629183943468, 2.987319184037793, 3.593896350074533, 4.032632801501248, 4.586945406780664, 5.288265027615739, 5.412231847248797, 6.008824186487559, 6.813780848045421, 7.199697577587415, 7.562886353420246, 8.191839130706668, 8.439979705290436, 8.960203935747749, 9.656754197780199, 9.892737197854014, 10.34719019491919, 10.79395602331628, 11.13275422995836, 11.77791327944854, 12.17905226505455, 12.71922845826293, 13.12746433424192
