| L(s) = 1 | − 3·3-s − 2·4-s + 5-s + 7-s + 6·9-s − 6·11-s + 6·12-s − 9·13-s − 3·15-s − 6·17-s − 2·20-s − 3·21-s − 12·23-s − 9·27-s − 2·28-s + 3·29-s − 6·31-s + 18·33-s + 35-s − 12·36-s + 20·37-s + 27·39-s − 12·41-s + 8·43-s + 12·44-s + 6·45-s − 6·49-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 4-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.80·11-s + 1.73·12-s − 2.49·13-s − 0.774·15-s − 1.45·17-s − 0.447·20-s − 0.654·21-s − 2.50·23-s − 1.73·27-s − 0.377·28-s + 0.557·29-s − 1.07·31-s + 3.13·33-s + 0.169·35-s − 2·36-s + 3.28·37-s + 4.32·39-s − 1.87·41-s + 1.21·43-s + 1.80·44-s + 0.894·45-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1114102972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1114102972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−11.86131838813587674782220722380, −11.39502071496164773830388195352, −11.05671761065448445732728550764, −10.36991115359669123604800749312, −10.11882182613055785969880292227, −9.573330021712830088844340274017, −9.567634371287389999357717214378, −8.514538950817179259346087612720, −7.943463777893820634078176771628, −7.66183269479663247546267361554, −6.96294748897275320710413980460, −6.47636871994357806988692688867, −5.75152316705375807202607760442, −5.37501960139993051197869037323, −4.93082024653125411351835608108, −4.47472094678161490026605803293, −4.15254855460881646394896294953, −2.48414006759323295599288425819, −2.17506326725106952223255566359, −0.25414839763193144939399781602,
0.25414839763193144939399781602, 2.17506326725106952223255566359, 2.48414006759323295599288425819, 4.15254855460881646394896294953, 4.47472094678161490026605803293, 4.93082024653125411351835608108, 5.37501960139993051197869037323, 5.75152316705375807202607760442, 6.47636871994357806988692688867, 6.96294748897275320710413980460, 7.66183269479663247546267361554, 7.943463777893820634078176771628, 8.514538950817179259346087612720, 9.567634371287389999357717214378, 9.573330021712830088844340274017, 10.11882182613055785969880292227, 10.36991115359669123604800749312, 11.05671761065448445732728550764, 11.39502071496164773830388195352, 11.86131838813587674782220722380
