| L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s − 12·17-s + 12·19-s − 10·25-s − 4·27-s − 16·33-s + 16·41-s + 24·43-s − 10·49-s + 24·51-s − 24·57-s + 4·67-s + 28·73-s + 20·75-s + 5·81-s + 16·83-s + 8·89-s + 28·97-s + 24·99-s − 8·107-s − 20·113-s + 26·121-s − 32·123-s + 127-s − 48·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s − 2.91·17-s + 2.75·19-s − 2·25-s − 0.769·27-s − 2.78·33-s + 2.49·41-s + 3.65·43-s − 1.42·49-s + 3.36·51-s − 3.17·57-s + 0.488·67-s + 3.27·73-s + 2.30·75-s + 5/9·81-s + 1.75·83-s + 0.847·89-s + 2.84·97-s + 2.41·99-s − 0.773·107-s − 1.88·113-s + 2.36·121-s − 2.88·123-s + 0.0887·127-s − 4.22·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3115008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3115008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012232477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012232477\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.26948453317897544910090398161, −7.16333848096912753102386607562, −6.57091938769509480942354725697, −6.26794152529161447517798404710, −5.98247083912518480972422452649, −5.59232004512023197121827602220, −4.97254580700680097900229302596, −4.48413639840570454203249914552, −4.13958008081477749813044275306, −3.84092207155881713644722109378, −3.29289095796707662296197413885, −2.22728463189050631057902889299, −2.06399639050261723146787386632, −1.02393888089349693967293608434, −0.73510269161810698907992625597,
0.73510269161810698907992625597, 1.02393888089349693967293608434, 2.06399639050261723146787386632, 2.22728463189050631057902889299, 3.29289095796707662296197413885, 3.84092207155881713644722109378, 4.13958008081477749813044275306, 4.48413639840570454203249914552, 4.97254580700680097900229302596, 5.59232004512023197121827602220, 5.98247083912518480972422452649, 6.26794152529161447517798404710, 6.57091938769509480942354725697, 7.16333848096912753102386607562, 7.26948453317897544910090398161
