| L(s) = 1 | + 2.74·3-s + 2.09·5-s + 3.15·7-s + 4.52·9-s + 4.30·11-s − 5.37·13-s + 5.73·15-s − 0.485·17-s + 2.21·19-s + 8.66·21-s − 6.89·23-s − 0.625·25-s + 4.17·27-s + 0.0910·29-s − 1.99·31-s + 11.7·33-s + 6.60·35-s + 11.1·37-s − 14.7·39-s + 10.2·41-s − 4.00·43-s + 9.45·45-s − 1.52·47-s + 2.98·49-s − 1.33·51-s + 2.30·53-s + 8.99·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + 0.935·5-s + 1.19·7-s + 1.50·9-s + 1.29·11-s − 1.49·13-s + 1.48·15-s − 0.117·17-s + 0.507·19-s + 1.89·21-s − 1.43·23-s − 0.125·25-s + 0.802·27-s + 0.0169·29-s − 0.358·31-s + 2.05·33-s + 1.11·35-s + 1.83·37-s − 2.36·39-s + 1.59·41-s − 0.611·43-s + 1.40·45-s − 0.222·47-s + 0.426·49-s − 0.186·51-s + 0.316·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.881448352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.881448352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 + 0.485T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 0.0910T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.00T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 6.37T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−8.449343902554955149518811419058, −7.77858514659782827580626508934, −7.30644199318189577870937234912, −6.28601753509383525436805958725, −5.41974202662520048984078531975, −4.43308287768877664842724506117, −3.89956920295934218174451591245, −2.64481285968925917821948896173, −2.12116936688054895173455487229, −1.36280695299262981537672038303,
1.36280695299262981537672038303, 2.12116936688054895173455487229, 2.64481285968925917821948896173, 3.89956920295934218174451591245, 4.43308287768877664842724506117, 5.41974202662520048984078531975, 6.28601753509383525436805958725, 7.30644199318189577870937234912, 7.77858514659782827580626508934, 8.449343902554955149518811419058
