L(s) = 1 | + 2·11-s − 12·19-s + 2·29-s − 4·31-s + 16·41-s − 49-s + 20·59-s + 18·71-s − 2·79-s + 4·89-s + 24·101-s − 18·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
L(s) = 1 | + 0.603·11-s − 2.75·19-s + 0.371·29-s − 0.718·31-s + 2.49·41-s − 1/7·49-s + 2.60·59-s + 2.13·71-s − 0.225·79-s + 0.423·89-s + 2.38·101-s − 1.72·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.445176074\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.445176074\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.218669316908950529251621970975, −7.934720996971580467658431262459, −7.51244722779132455488311938850, −7.12350791899728955739604573591, −6.64725579999132615347491504221, −6.57740782418473537831566035843, −6.10270164826179713246749413678, −5.86906162130743002115587692141, −5.35323659869098991917157244386, −4.99241485170495571128446592611, −4.52020567753394969491021155998, −4.16033409312504071078889926326, −3.82288345560584941836361164037, −3.68640903147524051846045871005, −2.83445810712559253034214697924, −2.49197133748881539672909304268, −2.10422136582708275422063904289, −1.72636298507664953735269052166, −0.920772175312508306128775001377, −0.46438957403768265157818910288,
0.46438957403768265157818910288, 0.920772175312508306128775001377, 1.72636298507664953735269052166, 2.10422136582708275422063904289, 2.49197133748881539672909304268, 2.83445810712559253034214697924, 3.68640903147524051846045871005, 3.82288345560584941836361164037, 4.16033409312504071078889926326, 4.52020567753394969491021155998, 4.99241485170495571128446592611, 5.35323659869098991917157244386, 5.86906162130743002115587692141, 6.10270164826179713246749413678, 6.57740782418473537831566035843, 6.64725579999132615347491504221, 7.12350791899728955739604573591, 7.51244722779132455488311938850, 7.934720996971580467658431262459, 8.218669316908950529251621970975