L(s) = 1 | + 2·3-s + 3·9-s − 8·11-s + 4·17-s − 8·19-s + 25-s + 4·27-s − 16·33-s + 4·41-s + 8·43-s + 2·49-s + 8·51-s − 16·57-s − 24·59-s − 24·67-s + 20·73-s + 2·75-s + 5·81-s − 8·83-s − 12·89-s − 28·97-s − 24·99-s + 24·107-s − 12·113-s + 26·121-s + 8·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 2.41·11-s + 0.970·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s − 2.78·33-s + 0.624·41-s + 1.21·43-s + 2/7·49-s + 1.12·51-s − 2.11·57-s − 3.12·59-s − 2.93·67-s + 2.34·73-s + 0.230·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 2.84·97-s − 2.41·99-s + 2.32·107-s − 1.12·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.234396477105587706249662636004, −7.75866687525969370360077670275, −7.73435767676089003755122228124, −7.26992741343389521999119886505, −6.39854886739801755961916412419, −6.14269520510095343624413187280, −5.33556450763905259522444606662, −5.11098268655570501466709405394, −4.23445324460423189498619158076, −4.10961287212778855632833672573, −2.94053502252576914533622368524, −2.91524621673910645918598375111, −2.28860875976801017534156160572, −1.47970352314852327908767006362, 0,
1.47970352314852327908767006362, 2.28860875976801017534156160572, 2.91524621673910645918598375111, 2.94053502252576914533622368524, 4.10961287212778855632833672573, 4.23445324460423189498619158076, 5.11098268655570501466709405394, 5.33556450763905259522444606662, 6.14269520510095343624413187280, 6.39854886739801755961916412419, 7.26992741343389521999119886505, 7.73435767676089003755122228124, 7.75866687525969370360077670275, 8.234396477105587706249662636004