| L(s) = 1 | + 2-s + 4-s + 8-s − 8·13-s + 16-s − 12·17-s − 10·25-s − 8·26-s + 12·29-s + 32-s − 12·34-s + 4·37-s − 12·41-s + 49-s − 10·50-s − 8·52-s − 12·53-s + 12·58-s + 16·61-s + 64-s − 12·68-s + 4·73-s + 4·74-s − 12·82-s + 12·89-s − 20·97-s + 98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2.21·13-s + 1/4·16-s − 2.91·17-s − 2·25-s − 1.56·26-s + 2.22·29-s + 0.176·32-s − 2.05·34-s + 0.657·37-s − 1.87·41-s + 1/7·49-s − 1.41·50-s − 1.10·52-s − 1.64·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s − 1.45·68-s + 0.468·73-s + 0.464·74-s − 1.32·82-s + 1.27·89-s − 2.03·97-s + 0.101·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.207333736191082013174831123730, −8.686268182491541384374530740513, −7.967527986115067076241988284772, −7.80365099617056766338064266882, −6.84458804347565715458286520604, −6.71310127772875365703449938682, −6.34752255234086455771568318662, −5.31921059379060528260176389250, −5.05202336643031434355439952711, −4.32309265725798888275246837851, −4.19102102405251164216473944262, −3.04185374086143547449059084416, −2.39101862347891857041372548490, −1.98782431119292345428047370492, 0,
1.98782431119292345428047370492, 2.39101862347891857041372548490, 3.04185374086143547449059084416, 4.19102102405251164216473944262, 4.32309265725798888275246837851, 5.05202336643031434355439952711, 5.31921059379060528260176389250, 6.34752255234086455771568318662, 6.71310127772875365703449938682, 6.84458804347565715458286520604, 7.80365099617056766338064266882, 7.967527986115067076241988284772, 8.686268182491541384374530740513, 9.207333736191082013174831123730
