| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 5·9-s + 2·10-s − 5·11-s + 4·13-s + 16-s − 5·17-s − 5·18-s − 5·19-s + 2·20-s − 5·22-s + 10·23-s − 25-s + 4·26-s + 32-s − 5·34-s − 5·36-s + 9·37-s − 5·38-s + 2·40-s − 5·44-s − 10·45-s + 10·46-s + 10·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 5/3·9-s + 0.632·10-s − 1.50·11-s + 1.10·13-s + 1/4·16-s − 1.21·17-s − 1.17·18-s − 1.14·19-s + 0.447·20-s − 1.06·22-s + 2.08·23-s − 1/5·25-s + 0.784·26-s + 0.176·32-s − 0.857·34-s − 5/6·36-s + 1.47·37-s − 0.811·38-s + 0.316·40-s − 0.753·44-s − 1.49·45-s + 1.47·46-s + 10/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.690691788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690691788\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 170 T^{2} + p^{2} T^{4} \) |
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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
−8.492336978829953026520466228567, −8.035586408751654110469652061173, −7.64791646569346460586478633013, −6.87627969783762347910928055543, −6.46796304996739428931435231301, −6.10610369765021072568123110118, −5.71551432890070618711992134228, −5.11604660689576680420305942249, −5.00495782937041374641078164791, −4.16879638878826673791512279611, −3.57173253019391291569431744188, −2.84690576720299293677237153014, −2.47526744211796281546010497613, −2.07464116870629392684308581672, −0.73033646217119766230883916812,
0.73033646217119766230883916812, 2.07464116870629392684308581672, 2.47526744211796281546010497613, 2.84690576720299293677237153014, 3.57173253019391291569431744188, 4.16879638878826673791512279611, 5.00495782937041374641078164791, 5.11604660689576680420305942249, 5.71551432890070618711992134228, 6.10610369765021072568123110118, 6.46796304996739428931435231301, 6.87627969783762347910928055543, 7.64791646569346460586478633013, 8.035586408751654110469652061173, 8.492336978829953026520466228567
