| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s + 5·7-s − 8-s + 10-s + 4·11-s − 5·13-s + 5·14-s + 15-s − 16-s − 2·17-s + 5·19-s + 5·21-s + 4·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s + 4·33-s − 2·34-s + 5·35-s + 37-s + 5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.38·13-s + 1.33·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 1.14·19-s + 1.09·21-s + 0.852·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s + 0.696·33-s − 0.342·34-s + 0.845·35-s + 0.164·37-s + 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.251390660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.251390660\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−9.831343566594205917664549654888, −9.387715447092821216452960063542, −9.350636411152087656773196620287, −8.991774535737113864566293787271, −8.368236106500410927382404112243, −7.78069348500159354035072346839, −7.67836108611901298635361204773, −7.32663558808767185092253249957, −6.67289225518721656527527585044, −6.19557285676697858922175379338, −5.50005982686468226758144248732, −5.40896512500361743712494875341, −4.81426163624664657202230537606, −4.44348654643401444257823562430, −3.91568696558912322202087051379, −3.56268178767474618850704171106, −2.64771732596061523796186109014, −2.22617306963070611371022699356, −1.75047599663205583719936389376, −0.921361052193918108075689182150,
0.921361052193918108075689182150, 1.75047599663205583719936389376, 2.22617306963070611371022699356, 2.64771732596061523796186109014, 3.56268178767474618850704171106, 3.91568696558912322202087051379, 4.44348654643401444257823562430, 4.81426163624664657202230537606, 5.40896512500361743712494875341, 5.50005982686468226758144248732, 6.19557285676697858922175379338, 6.67289225518721656527527585044, 7.32663558808767185092253249957, 7.67836108611901298635361204773, 7.78069348500159354035072346839, 8.368236106500410927382404112243, 8.991774535737113864566293787271, 9.350636411152087656773196620287, 9.387715447092821216452960063542, 9.831343566594205917664549654888
