| L(s) = 1 | − 4-s − 11-s − 3·16-s + 25-s − 14·31-s + 16·37-s + 44-s − 6·47-s − 4·49-s − 6·53-s + 12·59-s + 7·64-s − 2·67-s + 6·71-s − 18·89-s + 4·97-s − 100-s + 16·103-s + 12·113-s + 121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.301·11-s − 3/4·16-s + 1/5·25-s − 2.51·31-s + 2.63·37-s + 0.150·44-s − 0.875·47-s − 4/7·49-s − 0.824·53-s + 1.56·59-s + 7/8·64-s − 0.244·67-s + 0.712·71-s − 1.90·89-s + 0.406·97-s − 0.0999·100-s + 1.57·103-s + 1.12·113-s + 1/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.36828250900503417275095834972, −7.08427307801830795641955571937, −6.56770762327839215421996821253, −6.10544281123673679328718070451, −5.68752923116124310684381104274, −5.28337990826132128961816141009, −4.74340941157951287670358044241, −4.45701172528653151946818565300, −3.91174842651292112836066981197, −3.44997528715722504141228418169, −2.89616505606515762031794926470, −2.25423524075303761594945182227, −1.80090193517497396536391751852, −0.870024250421061824697470711606, 0,
0.870024250421061824697470711606, 1.80090193517497396536391751852, 2.25423524075303761594945182227, 2.89616505606515762031794926470, 3.44997528715722504141228418169, 3.91174842651292112836066981197, 4.45701172528653151946818565300, 4.74340941157951287670358044241, 5.28337990826132128961816141009, 5.68752923116124310684381104274, 6.10544281123673679328718070451, 6.56770762327839215421996821253, 7.08427307801830795641955571937, 7.36828250900503417275095834972
