L(s) = 1 | + 0.0962·3-s − 1.48·5-s − 7-s − 2.99·9-s − 11-s − 13-s − 0.143·15-s − 7.53·17-s + 7.59·19-s − 0.0962·21-s − 3.71·23-s − 2.78·25-s − 0.576·27-s − 9.29·29-s + 6.27·31-s − 0.0962·33-s + 1.48·35-s + 0.976·37-s − 0.0962·39-s + 4.24·41-s + 11.8·43-s + 4.44·45-s + 12.9·47-s + 49-s − 0.725·51-s − 4.71·53-s + 1.48·55-s + ⋯ |
L(s) = 1 | + 0.0555·3-s − 0.664·5-s − 0.377·7-s − 0.996·9-s − 0.301·11-s − 0.277·13-s − 0.0369·15-s − 1.82·17-s + 1.74·19-s − 0.0210·21-s − 0.775·23-s − 0.557·25-s − 0.110·27-s − 1.72·29-s + 1.12·31-s − 0.0167·33-s + 0.251·35-s + 0.160·37-s − 0.0154·39-s + 0.662·41-s + 1.81·43-s + 0.662·45-s + 1.88·47-s + 0.142·49-s − 0.101·51-s − 0.648·53-s + 0.200·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8803946670\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8803946670\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.0962T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 - 7.59T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 0.976T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 + 5.01T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 + 9.99T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 9.20T + 97T^{2} \) |
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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
−8.405262058766613277080694153114, −7.64228244583136364433109462957, −7.21568689721905653050509523476, −6.07571809416931066782626888975, −5.63904454724782136296871449610, −4.56136052226701392980220233384, −3.86023988242690875911140821139, −2.92873904115897717827762287804, −2.18343151295769478353011713357, −0.50556033794814371888545962170,
0.50556033794814371888545962170, 2.18343151295769478353011713357, 2.92873904115897717827762287804, 3.86023988242690875911140821139, 4.56136052226701392980220233384, 5.63904454724782136296871449610, 6.07571809416931066782626888975, 7.21568689721905653050509523476, 7.64228244583136364433109462957, 8.405262058766613277080694153114