| L(s) = 1 | − 2·3-s + 4·5-s + 4·7-s + 2·9-s + 2·11-s + 4·13-s − 8·15-s − 8·17-s − 8·21-s − 8·23-s + 8·25-s − 6·27-s + 12·29-s − 12·31-s − 4·33-s + 16·35-s − 16·37-s − 8·39-s + 2·41-s + 8·45-s + 8·49-s + 16·51-s + 8·55-s − 8·61-s + 8·63-s + 16·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1.51·7-s + 2/3·9-s + 0.603·11-s + 1.10·13-s − 2.06·15-s − 1.94·17-s − 1.74·21-s − 1.66·23-s + 8/5·25-s − 1.15·27-s + 2.22·29-s − 2.15·31-s − 0.696·33-s + 2.70·35-s − 2.63·37-s − 1.28·39-s + 0.312·41-s + 1.19·45-s + 8/7·49-s + 2.24·51-s + 1.07·55-s − 1.02·61-s + 1.00·63-s + 1.98·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.205480589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.205480589\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−13.31143634827532823143007410294, −13.28533952118508372847212185166, −12.17041539856394843093596484185, −12.04853780772347251702904442972, −11.24876756436953267215165874930, −11.03109013000032081372752797166, −10.38283417544277296419306950370, −10.23089008785348678323412894650, −9.100900418652268888806760474411, −9.010762990122238157370822239586, −8.341878165967709732250305975147, −7.52782824756656516030151549905, −6.52891240013108715853552633225, −6.44272941279775910592960789692, −5.70913527715080602288711552732, −5.24307603797811732552352746845, −4.60285408387943594428588493632, −3.80644469410721375548118373424, −2.02218370411221792134542555067, −1.69689616983828418430720355533,
1.69689616983828418430720355533, 2.02218370411221792134542555067, 3.80644469410721375548118373424, 4.60285408387943594428588493632, 5.24307603797811732552352746845, 5.70913527715080602288711552732, 6.44272941279775910592960789692, 6.52891240013108715853552633225, 7.52782824756656516030151549905, 8.341878165967709732250305975147, 9.010762990122238157370822239586, 9.100900418652268888806760474411, 10.23089008785348678323412894650, 10.38283417544277296419306950370, 11.03109013000032081372752797166, 11.24876756436953267215165874930, 12.04853780772347251702904442972, 12.17041539856394843093596484185, 13.28533952118508372847212185166, 13.31143634827532823143007410294
