| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s − 5-s + 6·6-s − 7-s − 4·8-s + 4·9-s + 2·10-s + 2·11-s − 9·12-s − 3·13-s + 2·14-s + 3·15-s + 5·16-s − 2·17-s − 8·18-s + 2·19-s − 3·20-s + 3·21-s − 4·22-s − 10·23-s + 12·24-s − 6·25-s + 6·26-s − 6·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.447·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 4/3·9-s + 0.632·10-s + 0.603·11-s − 2.59·12-s − 0.832·13-s + 0.534·14-s + 0.774·15-s + 5/4·16-s − 0.485·17-s − 1.88·18-s + 0.458·19-s − 0.670·20-s + 0.654·21-s − 0.852·22-s − 2.08·23-s + 2.44·24-s − 6/5·25-s + 1.17·26-s − 1.15·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 99 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 142 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 59 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 23 T + 271 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 319 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 166 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−11.06182501279895325052782927882, −10.60287725259275182621730642620, −9.986012943794927514747100801308, −9.635273451298823125456534079198, −9.407947924227090552040224529809, −8.892517283374611848571406957455, −7.951483734971078151639838276401, −7.76961005975310623714196819576, −7.41936718711275481373901664616, −6.72763010216460240577597558624, −6.15915467907456406136168820306, −6.06357461214933637977396780043, −5.37952484435793036848315259052, −4.82792745568016670431215353549, −3.83530467209852321084430827197, −3.56971673834977105145940364136, −2.18198330808258992032302602749, −1.65980106514964114310882374667, 0, 0,
1.65980106514964114310882374667, 2.18198330808258992032302602749, 3.56971673834977105145940364136, 3.83530467209852321084430827197, 4.82792745568016670431215353549, 5.37952484435793036848315259052, 6.06357461214933637977396780043, 6.15915467907456406136168820306, 6.72763010216460240577597558624, 7.41936718711275481373901664616, 7.76961005975310623714196819576, 7.951483734971078151639838276401, 8.892517283374611848571406957455, 9.407947924227090552040224529809, 9.635273451298823125456534079198, 9.986012943794927514747100801308, 10.60287725259275182621730642620, 11.06182501279895325052782927882
