| L(s) = 1 | + 2·2-s + 2·4-s + 2·11-s − 4·16-s − 2·17-s + 4·22-s − 4·25-s − 8·32-s − 4·34-s − 8·43-s + 4·44-s + 2·49-s − 8·50-s + 8·59-s − 8·64-s − 24·67-s − 4·68-s + 24·73-s + 10·83-s − 16·86-s − 12·89-s + 8·97-s + 4·98-s − 8·100-s + 8·107-s − 24·113-s + 16·118-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.603·11-s − 16-s − 0.485·17-s + 0.852·22-s − 4/5·25-s − 1.41·32-s − 0.685·34-s − 1.21·43-s + 0.603·44-s + 2/7·49-s − 1.13·50-s + 1.04·59-s − 64-s − 2.93·67-s − 0.485·68-s + 2.80·73-s + 1.09·83-s − 1.72·86-s − 1.27·89-s + 0.812·97-s + 0.404·98-s − 4/5·100-s + 0.773·107-s − 2.25·113-s + 1.47·118-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−7.53243719462398369502764904801, −6.83990258784341316074220462461, −6.62168632306244290624907758917, −6.33070555516512253953164461715, −5.62602015794669622386672951983, −5.50604866298035235565791843205, −4.89349708381494403846926613110, −4.40995796815223609766286640449, −4.13796587905890589690788253032, −3.45529799913797980791646480045, −3.30808100595079886139111177263, −2.43062931876858444043320087907, −2.08526944692380538505145998015, −1.23616753540325629750333318968, 0,
1.23616753540325629750333318968, 2.08526944692380538505145998015, 2.43062931876858444043320087907, 3.30808100595079886139111177263, 3.45529799913797980791646480045, 4.13796587905890589690788253032, 4.40995796815223609766286640449, 4.89349708381494403846926613110, 5.50604866298035235565791843205, 5.62602015794669622386672951983, 6.33070555516512253953164461715, 6.62168632306244290624907758917, 6.83990258784341316074220462461, 7.53243719462398369502764904801
