| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 5·7-s + 4·8-s + 3·9-s − 4·10-s − 6·12-s + 13-s − 10·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s − 19-s − 6·20-s + 10·21-s + 3·23-s − 8·24-s + 3·25-s + 2·26-s − 4·27-s − 15·28-s + 6·29-s + 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.88·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.73·12-s + 0.277·13-s − 2.67·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s − 1.34·20-s + 2.18·21-s + 0.625·23-s − 1.63·24-s + 3/5·25-s + 0.392·26-s − 0.769·27-s − 2.83·28-s + 1.11·29-s + 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 37 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 95 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 27 T + 299 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 146 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 239 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.131396103738664002563947691408, −7.75990508175538024717433768943, −7.38031093531554931148070203961, −7.15430403376256866444549091850, −6.54602188916894816412668355635, −6.54461564772389815547244424725, −6.01230982118239416469352634948, −5.85543514909425100242439875061, −5.26944688824655288825594300211, −5.04907235577840709778785839998, −4.38449580434776504274105986292, −4.29945039865810197247479018627, −3.59920611396912255574573407326, −3.41365641417666246485100237491, −2.95154696503534740493843135233, −2.72726370297914536668279575071, −1.48036518583715417267933486074, −1.40015458791826212768109164545, 0, 0,
1.40015458791826212768109164545, 1.48036518583715417267933486074, 2.72726370297914536668279575071, 2.95154696503534740493843135233, 3.41365641417666246485100237491, 3.59920611396912255574573407326, 4.29945039865810197247479018627, 4.38449580434776504274105986292, 5.04907235577840709778785839998, 5.26944688824655288825594300211, 5.85543514909425100242439875061, 6.01230982118239416469352634948, 6.54461564772389815547244424725, 6.54602188916894816412668355635, 7.15430403376256866444549091850, 7.38031093531554931148070203961, 7.75990508175538024717433768943, 8.131396103738664002563947691408
