| L(s) = 1 | + 2·3-s − 4·5-s − 6·7-s + 2·9-s + 2·11-s + 8·13-s − 8·15-s − 12·21-s + 2·23-s + 8·25-s + 6·27-s − 4·29-s + 6·31-s + 4·33-s + 24·35-s − 12·37-s + 16·39-s + 16·41-s − 8·45-s + 24·47-s + 18·49-s − 8·55-s − 12·61-s − 12·63-s − 32·65-s − 8·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 2.26·7-s + 2/3·9-s + 0.603·11-s + 2.21·13-s − 2.06·15-s − 2.61·21-s + 0.417·23-s + 8/5·25-s + 1.15·27-s − 0.742·29-s + 1.07·31-s + 0.696·33-s + 4.05·35-s − 1.97·37-s + 2.56·39-s + 2.49·41-s − 1.19·45-s + 3.50·47-s + 18/7·49-s − 1.07·55-s − 1.53·61-s − 1.51·63-s − 3.96·65-s − 0.977·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754588269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754588269\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−10.02038568606393992027003324912, −9.274183496705834227785873470824, −9.033451179060103366143957247892, −8.803727793630273977948568625812, −8.606798756697154697149439978722, −7.940728905914423819862839502081, −7.51843790364499834845475554688, −7.22801875838778529820540460906, −6.76112498216717256554361702143, −6.22691326441668378471835840796, −6.08210519260804003265035426469, −5.41983547877019168194841823456, −4.36172869450521989405388897874, −4.14393057123534156066272538732, −3.60312896463530937586643056678, −3.55504923694438526970331634079, −2.91371816638739528541907309216, −2.59002122685879342603562033705, −1.29236337608698125213433333438, −0.60076762091017680737053326228,
0.60076762091017680737053326228, 1.29236337608698125213433333438, 2.59002122685879342603562033705, 2.91371816638739528541907309216, 3.55504923694438526970331634079, 3.60312896463530937586643056678, 4.14393057123534156066272538732, 4.36172869450521989405388897874, 5.41983547877019168194841823456, 6.08210519260804003265035426469, 6.22691326441668378471835840796, 6.76112498216717256554361702143, 7.22801875838778529820540460906, 7.51843790364499834845475554688, 7.940728905914423819862839502081, 8.606798756697154697149439978722, 8.803727793630273977948568625812, 9.033451179060103366143957247892, 9.274183496705834227785873470824, 10.02038568606393992027003324912
