| L(s) = 1 | + 2·2-s + 3·4-s − 5-s − 7-s + 4·8-s − 9-s − 2·10-s + 13-s − 2·14-s + 5·16-s − 8·17-s − 2·18-s + 2·19-s − 3·20-s + 4·23-s − 8·25-s + 2·26-s − 3·28-s − 12·29-s − 8·31-s + 6·32-s − 16·34-s + 35-s − 3·36-s − 9·37-s + 4·38-s − 4·40-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.447·5-s − 0.377·7-s + 1.41·8-s − 1/3·9-s − 0.632·10-s + 0.277·13-s − 0.534·14-s + 5/4·16-s − 1.94·17-s − 0.471·18-s + 0.458·19-s − 0.670·20-s + 0.834·23-s − 8/5·25-s + 0.392·26-s − 0.566·28-s − 2.22·29-s − 1.43·31-s + 1.06·32-s − 2.74·34-s + 0.169·35-s − 1/2·36-s − 1.47·37-s + 0.648·38-s − 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 143 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 151 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 155 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.938311287615067027256450119284, −7.57241471340565279441833351710, −7.19067486828980283723250443888, −7.18980898680577379130674501176, −6.64884586895535107843347327122, −6.13418160963120084366446820015, −5.80450665565625033131947837239, −5.72856354291062477776001609055, −5.13084967746220823056961227858, −4.74843159085696755802413995431, −4.37548146909073407953712513346, −3.93915775758761204684414141910, −3.67542164100473162127555079968, −3.16863260806119009757599960044, −2.97583869090126255660477465309, −2.21750881400812327202680498540, −1.77857402604288117919910419113, −1.53786153383204316574314039475, 0, 0,
1.53786153383204316574314039475, 1.77857402604288117919910419113, 2.21750881400812327202680498540, 2.97583869090126255660477465309, 3.16863260806119009757599960044, 3.67542164100473162127555079968, 3.93915775758761204684414141910, 4.37548146909073407953712513346, 4.74843159085696755802413995431, 5.13084967746220823056961227858, 5.72856354291062477776001609055, 5.80450665565625033131947837239, 6.13418160963120084366446820015, 6.64884586895535107843347327122, 7.18980898680577379130674501176, 7.19067486828980283723250443888, 7.57241471340565279441833351710, 7.938311287615067027256450119284
