| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s + 2·13-s + 8·19-s + 6·23-s − 7·25-s − 10·29-s + 10·31-s + 4·35-s + 8·37-s + 14·41-s + 10·43-s − 2·47-s − 5·49-s − 8·53-s + 4·55-s − 14·59-s − 6·61-s + 4·65-s + 10·67-s + 4·71-s + 4·77-s + 22·79-s + 6·83-s + 16·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 1.83·19-s + 1.25·23-s − 7/5·25-s − 1.85·29-s + 1.79·31-s + 0.676·35-s + 1.31·37-s + 2.18·41-s + 1.52·43-s − 0.291·47-s − 5/7·49-s − 1.09·53-s + 0.539·55-s − 1.82·59-s − 0.768·61-s + 0.496·65-s + 1.22·67-s + 0.474·71-s + 0.455·77-s + 2.47·79-s + 0.658·83-s + 1.69·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.877155494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.877155494\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 81 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 113 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 105 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 273 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 171 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.241744644109786550504293156999, −8.929051590057102625231748723120, −8.081242119322877812522405451933, −7.899487035547687580181805581581, −7.54650527861848958411075158353, −7.48079566639795081378120836513, −6.47709954133768473051567214354, −6.46165624599156787413553147635, −5.85990200018754436615653833669, −5.78577968604400000709266472503, −5.04984282601921882511924223682, −4.94456848581861923647644462683, −4.23222026460836029514839551759, −4.00156388926641363270378352749, −3.17482802772969240542456665036, −3.10259376009666129090251951922, −2.14083260100744734223632123731, −1.98753436473148648330870904532, −1.09291123114302028547127821993, −0.900747617632598009084088353306,
0.900747617632598009084088353306, 1.09291123114302028547127821993, 1.98753436473148648330870904532, 2.14083260100744734223632123731, 3.10259376009666129090251951922, 3.17482802772969240542456665036, 4.00156388926641363270378352749, 4.23222026460836029514839551759, 4.94456848581861923647644462683, 5.04984282601921882511924223682, 5.78577968604400000709266472503, 5.85990200018754436615653833669, 6.46165624599156787413553147635, 6.47709954133768473051567214354, 7.48079566639795081378120836513, 7.54650527861848958411075158353, 7.899487035547687580181805581581, 8.081242119322877812522405451933, 8.929051590057102625231748723120, 9.241744644109786550504293156999
