| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 4·5-s + 4·6-s + 4·8-s + 3·9-s − 8·10-s − 2·11-s + 6·12-s − 4·13-s − 8·15-s + 5·16-s − 6·17-s + 6·18-s + 2·19-s − 12·20-s − 4·22-s − 2·23-s + 8·24-s + 5·25-s − 8·26-s + 4·27-s + 2·29-s − 16·30-s − 2·31-s + 6·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s + 1.41·8-s + 9-s − 2.52·10-s − 0.603·11-s + 1.73·12-s − 1.10·13-s − 2.06·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 0.458·19-s − 2.68·20-s − 0.852·22-s − 0.417·23-s + 1.63·24-s + 25-s − 1.56·26-s + 0.769·27-s + 0.371·29-s − 2.92·30-s − 0.359·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 128 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 164 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 207 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.87161715423354416917858541009, −7.72718546031699091836186171033, −7.09493970938906417932740568874, −7.00951000310210907301621111008, −6.63448965681068405462370488281, −6.38562357619285748485463952781, −5.52978500443578023790699176274, −5.34625798815011729364805874231, −4.77426864258000334828983215385, −4.64965312842161796645698201590, −4.26001765418966900283550620124, −3.83989765421421113085205155432, −3.53169997269261568284827693797, −3.17851505547891205107054232439, −2.68718870221100730654388328463, −2.51027808792169283170464127427, −1.74966953783087070367555840812, −1.48567390407238931101725687018, 0, 0,
1.48567390407238931101725687018, 1.74966953783087070367555840812, 2.51027808792169283170464127427, 2.68718870221100730654388328463, 3.17851505547891205107054232439, 3.53169997269261568284827693797, 3.83989765421421113085205155432, 4.26001765418966900283550620124, 4.64965312842161796645698201590, 4.77426864258000334828983215385, 5.34625798815011729364805874231, 5.52978500443578023790699176274, 6.38562357619285748485463952781, 6.63448965681068405462370488281, 7.00951000310210907301621111008, 7.09493970938906417932740568874, 7.72718546031699091836186171033, 7.87161715423354416917858541009
