L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s − 4·17-s − 16·19-s + 25-s − 4·27-s + 16·33-s − 12·41-s − 8·43-s − 14·49-s + 8·51-s + 32·57-s − 24·59-s + 24·67-s + 4·73-s − 2·75-s + 5·81-s + 8·83-s + 4·89-s − 28·97-s − 24·99-s + 24·107-s + 12·113-s + 26·121-s + 24·123-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s − 0.970·17-s − 3.67·19-s + 1/5·25-s − 0.769·27-s + 2.78·33-s − 1.87·41-s − 1.21·43-s − 2·49-s + 1.12·51-s + 4.23·57-s − 3.12·59-s + 2.93·67-s + 0.468·73-s − 0.230·75-s + 5/9·81-s + 0.878·83-s + 0.423·89-s − 2.84·97-s − 2.41·99-s + 2.32·107-s + 1.12·113-s + 2.36·121-s + 2.16·123-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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−7.985644465490789105614221217605, −7.888892617242428206551037061144, −6.84594392053537268258202260884, −6.76366111244607068879792174831, −6.25812475404630879226037514190, −5.86487938441035862983502527784, −5.10335741728591181213017059039, −4.78210199118180492802537017050, −4.59744237493557402533707286052, −3.76484405045814108002996889467, −3.01325599841700254495812174327, −2.15654793577695836540091578775, −1.90410956588849953091073290916, 0, 0,
1.90410956588849953091073290916, 2.15654793577695836540091578775, 3.01325599841700254495812174327, 3.76484405045814108002996889467, 4.59744237493557402533707286052, 4.78210199118180492802537017050, 5.10335741728591181213017059039, 5.86487938441035862983502527784, 6.25812475404630879226037514190, 6.76366111244607068879792174831, 6.84594392053537268258202260884, 7.888892617242428206551037061144, 7.985644465490789105614221217605