| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 5·9-s − 12·11-s + 5·16-s + 10·18-s + 24·22-s − 6·23-s + 25-s − 6·29-s − 6·32-s − 15·36-s − 8·37-s − 14·43-s − 36·44-s + 12·46-s − 2·50-s − 12·53-s + 12·58-s + 7·64-s + 10·67-s − 12·71-s + 20·72-s + 16·74-s + 4·79-s + 16·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 5/3·9-s − 3.61·11-s + 5/4·16-s + 2.35·18-s + 5.11·22-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.06·32-s − 5/2·36-s − 1.31·37-s − 2.13·43-s − 5.42·44-s + 1.76·46-s − 0.282·50-s − 1.64·53-s + 1.57·58-s + 7/8·64-s + 1.22·67-s − 1.42·71-s + 2.35·72-s + 1.85·74-s + 0.450·79-s + 16/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.561693841321835015066175400037, −8.031786699570847412544851462124, −7.891065010743753563141671745228, −7.32537691646495858134302885194, −6.80099627830459230616857400983, −5.86045038904652423021831371411, −5.81363499888832150390624638766, −5.17114383975534663234239403824, −4.80934091914108181351418031433, −3.33073330941824489356906036908, −3.14836793146993122992887929389, −2.34751690342958102464014612548, −1.99263767242192126821507447658, 0, 0,
1.99263767242192126821507447658, 2.34751690342958102464014612548, 3.14836793146993122992887929389, 3.33073330941824489356906036908, 4.80934091914108181351418031433, 5.17114383975534663234239403824, 5.81363499888832150390624638766, 5.86045038904652423021831371411, 6.80099627830459230616857400983, 7.32537691646495858134302885194, 7.891065010743753563141671745228, 8.031786699570847412544851462124, 8.561693841321835015066175400037
