L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s + 2·13-s + 14-s − 16-s + 2·17-s + 4·19-s − 2·20-s − 25-s − 2·26-s + 28-s + 29-s − 8·31-s − 5·32-s − 2·34-s − 2·35-s − 10·37-s − 4·38-s + 6·40-s − 6·41-s − 12·43-s + 8·47-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.185·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s − 1.64·37-s − 0.648·38-s + 0.948·40-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9389620303\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9389620303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06393181021712, −12.56956131361046, −12.01938416884210, −11.58438560186048, −10.81695719543798, −10.51752023996008, −10.10763000430194, −9.537489968398973, −9.371287210053131, −8.816584640943809, −8.366069453481992, −7.854846138152870, −7.307720193273748, −6.842198827474569, −6.289737605355255, −5.636948576421212, −5.274519895828632, −4.893228850780636, −4.045454541889537, −3.471118402192117, −3.187695161619867, −2.150232119159776, −1.649401491917834, −1.217577690466507, −0.3188949417562131,
0.3188949417562131, 1.217577690466507, 1.649401491917834, 2.150232119159776, 3.187695161619867, 3.471118402192117, 4.045454541889537, 4.893228850780636, 5.274519895828632, 5.636948576421212, 6.289737605355255, 6.842198827474569, 7.307720193273748, 7.854846138152870, 8.366069453481992, 8.816584640943809, 9.371287210053131, 9.537489968398973, 10.10763000430194, 10.51752023996008, 10.81695719543798, 11.58438560186048, 12.01938416884210, 12.56956131361046, 13.06393181021712