| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 4·11-s − 4·13-s + 5·16-s + 4·17-s − 8·19-s − 6·20-s + 8·22-s + 3·25-s − 8·26-s − 4·29-s − 12·31-s + 6·32-s + 8·34-s − 12·37-s − 16·38-s − 8·40-s + 4·41-s + 12·44-s − 8·47-s − 14·49-s + 6·50-s − 12·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 1.20·11-s − 1.10·13-s + 5/4·16-s + 0.970·17-s − 1.83·19-s − 1.34·20-s + 1.70·22-s + 3/5·25-s − 1.56·26-s − 0.742·29-s − 2.15·31-s + 1.06·32-s + 1.37·34-s − 1.97·37-s − 2.59·38-s − 1.26·40-s + 0.624·41-s + 1.80·44-s − 1.16·47-s − 2·49-s + 0.848·50-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76212900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76212900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 140 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.35530157221326996685943682920, −7.18298818560264334881602170774, −6.78651954491416157817236833423, −6.74072327988054451652344052864, −6.03115136505126364773691844036, −5.93597785883246900198405627944, −5.42163226938694313670750874439, −5.05248737899353087565479658848, −4.85378909865608990849894213912, −4.27990149814969329738596891696, −4.11306795943653133354514350508, −3.71533763413300632751657607448, −3.31684282255202214840746155227, −3.25322810653105745612534491741, −2.33810130824716587787282626748, −2.25649922777307867798635589854, −1.46017927007483703242819705634, −1.40628023808543748823630715330, 0, 0,
1.40628023808543748823630715330, 1.46017927007483703242819705634, 2.25649922777307867798635589854, 2.33810130824716587787282626748, 3.25322810653105745612534491741, 3.31684282255202214840746155227, 3.71533763413300632751657607448, 4.11306795943653133354514350508, 4.27990149814969329738596891696, 4.85378909865608990849894213912, 5.05248737899353087565479658848, 5.42163226938694313670750874439, 5.93597785883246900198405627944, 6.03115136505126364773691844036, 6.74072327988054451652344052864, 6.78651954491416157817236833423, 7.18298818560264334881602170774, 7.35530157221326996685943682920
