| L(s) = 1 | + 2·5-s + 5·7-s − 11-s − 2·13-s − 17-s + 2·19-s − 9·23-s + 3·25-s − 6·29-s − 12·31-s + 10·35-s − 9·37-s + 3·41-s + 2·43-s − 14·47-s + 9·49-s + 3·53-s − 2·55-s − 24·59-s − 61-s − 4·65-s − 2·67-s − 25·71-s − 12·73-s − 5·77-s + 3·79-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.88·7-s − 0.301·11-s − 0.554·13-s − 0.242·17-s + 0.458·19-s − 1.87·23-s + 3/5·25-s − 1.11·29-s − 2.15·31-s + 1.69·35-s − 1.47·37-s + 0.468·41-s + 0.304·43-s − 2.04·47-s + 9/7·49-s + 0.412·53-s − 0.269·55-s − 3.12·59-s − 0.128·61-s − 0.496·65-s − 0.244·67-s − 2.96·71-s − 1.40·73-s − 0.569·77-s + 0.337·79-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 160 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−7.63745624210096357171202597080, −7.53118532001094719175023556090, −6.70668270789967633890177593551, −6.63264688609698015462378773702, −6.00866554670626865042079744714, −5.79986410952386088974612850144, −5.37785113468469007359117471042, −5.25177300777133945870242579394, −4.77639799981178319614907430538, −4.49960807842746904731067276608, −4.16076445854621581259954130651, −3.61881409878745915715574603080, −3.22703814353017751886902231326, −2.79710430694435538976692478721, −2.07720068936321360482182710078, −1.97255672866968911267077175713, −1.48507205216074216520246176703, −1.43757561007400231991021636840, 0, 0,
1.43757561007400231991021636840, 1.48507205216074216520246176703, 1.97255672866968911267077175713, 2.07720068936321360482182710078, 2.79710430694435538976692478721, 3.22703814353017751886902231326, 3.61881409878745915715574603080, 4.16076445854621581259954130651, 4.49960807842746904731067276608, 4.77639799981178319614907430538, 5.25177300777133945870242579394, 5.37785113468469007359117471042, 5.79986410952386088974612850144, 6.00866554670626865042079744714, 6.63264688609698015462378773702, 6.70668270789967633890177593551, 7.53118532001094719175023556090, 7.63745624210096357171202597080
