L(s) = 1 | − 3-s + 9-s − 12·13-s − 8·19-s − 27-s − 16·31-s + 4·37-s + 12·39-s − 24·43-s − 14·49-s + 8·57-s + 28·61-s − 8·67-s + 12·73-s − 16·79-s + 81-s + 16·93-s − 4·97-s − 36·109-s − 4·111-s − 12·117-s − 6·121-s + 127-s + 24·129-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 3.32·13-s − 1.83·19-s − 0.192·27-s − 2.87·31-s + 0.657·37-s + 1.92·39-s − 3.65·43-s − 2·49-s + 1.05·57-s + 3.58·61-s − 0.977·67-s + 1.40·73-s − 1.80·79-s + 1/9·81-s + 1.65·93-s − 0.406·97-s − 3.44·109-s − 0.379·111-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 2.11·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.55727707153378484774141976097, −7.06475469847576410250018540799, −6.71920497811244846696035723021, −6.65963768785974737826480192145, −5.65229906256087841325445637569, −5.36683559095823513937522588289, −4.95639818252911756604562729940, −4.60954800977677454684167974744, −4.03205568616657235432062033061, −3.42683606015106566673715003849, −2.69982664061727220731686131413, −2.07933314479152275364533906758, −1.76966313536701547243535553599, 0, 0,
1.76966313536701547243535553599, 2.07933314479152275364533906758, 2.69982664061727220731686131413, 3.42683606015106566673715003849, 4.03205568616657235432062033061, 4.60954800977677454684167974744, 4.95639818252911756604562729940, 5.36683559095823513937522588289, 5.65229906256087841325445637569, 6.65963768785974737826480192145, 6.71920497811244846696035723021, 7.06475469847576410250018540799, 7.55727707153378484774141976097