| L(s) = 1 | + 8·7-s − 4·13-s + 16·19-s − 10·25-s + 8·31-s + 20·37-s + 16·43-s + 34·49-s − 28·61-s − 32·67-s − 20·73-s + 8·79-s − 32·91-s + 28·97-s − 40·103-s − 4·109-s − 22·121-s + 127-s + 131-s + 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1.10·13-s + 3.67·19-s − 2·25-s + 1.43·31-s + 3.28·37-s + 2.43·43-s + 34/7·49-s − 3.58·61-s − 3.90·67-s − 2.34·73-s + 0.900·79-s − 3.35·91-s + 2.84·97-s − 3.94·103-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 11.0·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.949171984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.949171984\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−13.36200740682740161086695658077, −12.06359362942184667385147107695, −12.06359362942184667385147107695, −11.66894852312690078300764483455, −11.66894852312690078300764483455, −10.82265776539167631942690570095, −10.82265776539167631942690570095, −9.825287305444944792800568324534, −9.825287305444944792800568324534, −9.003535964671978513824781543440, −9.003535964671978513824781543440, −7.73821550164192930578511356198, −7.73821550164192930578511356198, −7.59743334158084999547271626311, −7.59743334158084999547271626311, −6.01970007910918377722950045283, −6.01970007910918377722950045283, −5.09016984693454778890596036288, −5.09016984693454778890596036288, −4.25084195754076010426468082779, −4.25084195754076010426468082779, −2.73087009892762739420775730904, −2.73087009892762739420775730904, −1.33945102693755199909069874398, −1.33945102693755199909069874398,
1.33945102693755199909069874398, 1.33945102693755199909069874398, 2.73087009892762739420775730904, 2.73087009892762739420775730904, 4.25084195754076010426468082779, 4.25084195754076010426468082779, 5.09016984693454778890596036288, 5.09016984693454778890596036288, 6.01970007910918377722950045283, 6.01970007910918377722950045283, 7.59743334158084999547271626311, 7.59743334158084999547271626311, 7.73821550164192930578511356198, 7.73821550164192930578511356198, 9.003535964671978513824781543440, 9.003535964671978513824781543440, 9.825287305444944792800568324534, 9.825287305444944792800568324534, 10.82265776539167631942690570095, 10.82265776539167631942690570095, 11.66894852312690078300764483455, 11.66894852312690078300764483455, 12.06359362942184667385147107695, 12.06359362942184667385147107695, 13.36200740682740161086695658077
