| L(s) = 1 | + 2·9-s − 8·11-s − 16·19-s + 4·29-s + 8·31-s − 20·41-s + 10·49-s + 4·61-s − 24·71-s + 16·79-s − 5·81-s + 12·89-s − 16·99-s + 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 32·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.41·11-s − 3.67·19-s + 0.742·29-s + 1.43·31-s − 3.12·41-s + 10/7·49-s + 0.512·61-s − 2.84·71-s + 1.80·79-s − 5/9·81-s + 1.27·89-s − 1.60·99-s + 2.78·101-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9069448191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9069448191\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.47491908197939904203647127102, −10.21298246523348427992405062600, −9.985444267125944034037253921248, −9.054619824260205469458065305317, −8.610622263332691695908064414208, −8.255759889897118346353240409922, −8.232494828028282268116699260376, −7.37486174029341564514622938239, −7.15486157267252801260739776920, −6.41993009569643865772210702014, −6.26634532657254455606529633165, −5.64743794391825044208365965319, −4.94758215657426926633425028408, −4.63324212096648958181487935239, −4.34550743996633793885571959651, −3.49911211162865093505811948464, −2.88305794249530315726189983479, −2.19766779638084244796385281511, −1.94460540979308359566598400772, −0.44797271786889384386884608863,
0.44797271786889384386884608863, 1.94460540979308359566598400772, 2.19766779638084244796385281511, 2.88305794249530315726189983479, 3.49911211162865093505811948464, 4.34550743996633793885571959651, 4.63324212096648958181487935239, 4.94758215657426926633425028408, 5.64743794391825044208365965319, 6.26634532657254455606529633165, 6.41993009569643865772210702014, 7.15486157267252801260739776920, 7.37486174029341564514622938239, 8.232494828028282268116699260376, 8.255759889897118346353240409922, 8.610622263332691695908064414208, 9.054619824260205469458065305317, 9.985444267125944034037253921248, 10.21298246523348427992405062600, 10.47491908197939904203647127102
