| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s + 4·13-s + 2·15-s − 12·17-s − 8·19-s − 8·21-s + 25-s + 4·27-s + 12·29-s − 4·35-s + 4·37-s − 8·39-s − 45-s − 2·49-s + 24·51-s + 16·57-s + 4·63-s − 4·65-s − 24·71-s − 2·75-s − 11·81-s + 12·83-s + 12·85-s − 24·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 2.91·17-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s + 2.22·29-s − 0.676·35-s + 0.657·37-s − 1.28·39-s − 0.149·45-s − 2/7·49-s + 3.36·51-s + 2.11·57-s + 0.503·63-s − 0.496·65-s − 2.84·71-s − 0.230·75-s − 1.22·81-s + 1.31·83-s + 1.30·85-s − 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.552217550781204179646223792184, −8.414153815175170759478064426827, −7.75829788190596058655904575203, −7.04404877195059392381067414173, −6.57891116465648258947670054106, −6.19980862096884640149159428467, −6.00289914819194479789365648779, −4.81402812643904182698198570995, −4.78130792717525308450176413839, −4.45128384697109248786717154579, −3.81585178009759093780043742498, −2.70286396079966253678906788159, −2.09224015679654400014877541610, −1.22133419512922967350095870642, 0,
1.22133419512922967350095870642, 2.09224015679654400014877541610, 2.70286396079966253678906788159, 3.81585178009759093780043742498, 4.45128384697109248786717154579, 4.78130792717525308450176413839, 4.81402812643904182698198570995, 6.00289914819194479789365648779, 6.19980862096884640149159428467, 6.57891116465648258947670054106, 7.04404877195059392381067414173, 7.75829788190596058655904575203, 8.414153815175170759478064426827, 8.552217550781204179646223792184
