| L(s) = 1 | − 4-s − 2·5-s − 2·7-s − 6·11-s − 2·13-s − 3·16-s + 2·19-s + 2·20-s + 3·25-s + 2·28-s − 6·29-s + 4·31-s + 4·35-s − 2·37-s − 6·41-s − 2·43-s + 6·44-s − 8·49-s + 2·52-s + 12·53-s + 12·55-s − 12·59-s − 20·61-s + 7·64-s + 4·65-s + 16·67-s − 12·71-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 0.755·7-s − 1.80·11-s − 0.554·13-s − 3/4·16-s + 0.458·19-s + 0.447·20-s + 3/5·25-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s + 0.904·44-s − 8/7·49-s + 0.277·52-s + 1.64·53-s + 1.61·55-s − 1.56·59-s − 2.56·61-s + 7/8·64-s + 0.496·65-s + 1.95·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 256 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 168 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.883189261924100285367699278865, −9.662192409485576623441819539167, −8.940822958268431086479763387500, −8.918811508542577170579471553856, −8.081630085440141152762217752635, −7.925284616352481666971539413070, −7.51962887526028270450858107400, −6.96919026304098461403641387292, −6.68577435659982868662434597231, −5.98955303734139928413255293136, −5.31448895197748011351559390899, −5.22356174996129912622679127077, −4.42657179225334757009567689386, −4.26098337197706662680269410074, −3.38757928029076444734609280301, −2.97400430415512322974018382064, −2.55116332079524321306653284472, −1.55618517111585788553635139092, 0, 0,
1.55618517111585788553635139092, 2.55116332079524321306653284472, 2.97400430415512322974018382064, 3.38757928029076444734609280301, 4.26098337197706662680269410074, 4.42657179225334757009567689386, 5.22356174996129912622679127077, 5.31448895197748011351559390899, 5.98955303734139928413255293136, 6.68577435659982868662434597231, 6.96919026304098461403641387292, 7.51962887526028270450858107400, 7.925284616352481666971539413070, 8.081630085440141152762217752635, 8.918811508542577170579471553856, 8.940822958268431086479763387500, 9.662192409485576623441819539167, 9.883189261924100285367699278865
