| L(s) = 1 | − 2·3-s + 2·5-s − 3·7-s + 3·9-s − 3·11-s − 2·13-s − 4·15-s + 5·17-s + 6·21-s − 3·23-s + 3·25-s − 4·27-s + 4·31-s + 6·33-s − 6·35-s − 3·37-s + 4·39-s + 5·41-s + 4·43-s + 6·45-s − 20·47-s − 3·49-s − 10·51-s − 11·53-s − 6·55-s − 14·59-s − 9·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.13·7-s + 9-s − 0.904·11-s − 0.554·13-s − 1.03·15-s + 1.21·17-s + 1.30·21-s − 0.625·23-s + 3/5·25-s − 0.769·27-s + 0.718·31-s + 1.04·33-s − 1.01·35-s − 0.493·37-s + 0.640·39-s + 0.780·41-s + 0.609·43-s + 0.894·45-s − 2.91·47-s − 3/7·49-s − 1.40·51-s − 1.51·53-s − 0.809·55-s − 1.82·59-s − 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 84 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 132 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T - 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.295459862741191836366426716587, −8.096466248614060578383822796344, −7.73282602481244865352057937286, −7.38335680708902944724423216033, −6.76098437633260733894448116856, −6.55669611979176050120679373129, −6.09561478012970703556510083661, −6.05146883656682760941857500636, −5.35403284812277342177230930807, −5.26261330962145684771709161699, −4.70261506558385583493030752457, −4.44780095956228908891380623807, −3.70624957004007185929608477620, −3.16822961883214977475307276114, −2.91330545058928577880743304605, −2.37694868165228821202556739528, −1.52635681863937996679831414451, −1.30747236893188544551075296630, 0, 0,
1.30747236893188544551075296630, 1.52635681863937996679831414451, 2.37694868165228821202556739528, 2.91330545058928577880743304605, 3.16822961883214977475307276114, 3.70624957004007185929608477620, 4.44780095956228908891380623807, 4.70261506558385583493030752457, 5.26261330962145684771709161699, 5.35403284812277342177230930807, 6.05146883656682760941857500636, 6.09561478012970703556510083661, 6.55669611979176050120679373129, 6.76098437633260733894448116856, 7.38335680708902944724423216033, 7.73282602481244865352057937286, 8.096466248614060578383822796344, 8.295459862741191836366426716587
