| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 4·8-s + 3·9-s + 4·10-s − 6·12-s + 6·13-s + 4·15-s + 5·16-s − 6·18-s − 6·19-s − 6·20-s − 6·23-s + 8·24-s + 3·25-s − 12·26-s − 4·27-s − 6·29-s − 8·30-s + 2·31-s − 6·32-s + 9·36-s + 8·37-s + 12·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 1.73·12-s + 1.66·13-s + 1.03·15-s + 5/4·16-s − 1.41·18-s − 1.37·19-s − 1.34·20-s − 1.25·23-s + 1.63·24-s + 3/5·25-s − 2.35·26-s − 0.769·27-s − 1.11·29-s − 1.46·30-s + 0.359·31-s − 1.06·32-s + 3/2·36-s + 1.31·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 63 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 160 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 152 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 163 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 192 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.324844224770862820171153527596, −8.173919585377709649132198263372, −7.54948514793512619223710249766, −7.49226898522576709927089718009, −6.75842845898022563843528173810, −6.58107812354297730134851707571, −6.26072360778815300064599576220, −6.01695185334009713896981011473, −5.29878508865643172822866749359, −5.27227313942533191776816648756, −4.25773551370273983258489671745, −4.24448528402139759880970189596, −3.56981458265196184211152218128, −3.46374026646003030488158509477, −2.38930171308769237551043212125, −2.19216439962113769519622752861, −1.28663463415072569746770891007, −1.13302256105365421319875554345, 0, 0,
1.13302256105365421319875554345, 1.28663463415072569746770891007, 2.19216439962113769519622752861, 2.38930171308769237551043212125, 3.46374026646003030488158509477, 3.56981458265196184211152218128, 4.24448528402139759880970189596, 4.25773551370273983258489671745, 5.27227313942533191776816648756, 5.29878508865643172822866749359, 6.01695185334009713896981011473, 6.26072360778815300064599576220, 6.58107812354297730134851707571, 6.75842845898022563843528173810, 7.49226898522576709927089718009, 7.54948514793512619223710249766, 8.173919585377709649132198263372, 8.324844224770862820171153527596
