| L(s) = 1 | − 3·3-s + 2·5-s + 3·7-s + 3·9-s + 3·11-s + 2·13-s − 6·15-s + 7·17-s + 19-s − 9·21-s − 7·23-s + 3·25-s + 5·29-s + 8·31-s − 9·33-s + 6·35-s + 3·37-s − 6·39-s − 7·41-s − 9·43-s + 6·45-s − 16·47-s + 7·49-s − 21·51-s − 12·53-s + 6·55-s − 3·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 1.13·7-s + 9-s + 0.904·11-s + 0.554·13-s − 1.54·15-s + 1.69·17-s + 0.229·19-s − 1.96·21-s − 1.45·23-s + 3/5·25-s + 0.928·29-s + 1.43·31-s − 1.56·33-s + 1.01·35-s + 0.493·37-s − 0.960·39-s − 1.09·41-s − 1.37·43-s + 0.894·45-s − 2.33·47-s + 49-s − 2.94·51-s − 1.64·53-s + 0.809·55-s − 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.647503417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.647503417\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.15516183214452789590971172836, −9.851348328477962280274863918862, −9.651221308719995037582692903291, −8.794326596805942590056153439152, −8.395261910831623922141585940882, −8.130875103746570945378623723317, −7.70735351090577960309587627669, −6.98326482577137717952379021943, −6.41229212214052984800744432707, −6.35029149267896595404720225213, −5.89046720458883715911743777598, −5.33954242744985514883525626314, −5.19405892187006560771551621776, −4.62246804261990228396112262655, −4.17723899074870107867192432398, −3.37607624437404819989903804811, −2.89283832088833558021356842376, −1.66806304248655314634635255862, −1.57669554969915230220887290634, −0.70186031929410527275799333908,
0.70186031929410527275799333908, 1.57669554969915230220887290634, 1.66806304248655314634635255862, 2.89283832088833558021356842376, 3.37607624437404819989903804811, 4.17723899074870107867192432398, 4.62246804261990228396112262655, 5.19405892187006560771551621776, 5.33954242744985514883525626314, 5.89046720458883715911743777598, 6.35029149267896595404720225213, 6.41229212214052984800744432707, 6.98326482577137717952379021943, 7.70735351090577960309587627669, 8.130875103746570945378623723317, 8.395261910831623922141585940882, 8.794326596805942590056153439152, 9.651221308719995037582692903291, 9.851348328477962280274863918862, 10.15516183214452789590971172836
