| L(s) = 1 | + 3.27·3-s − 1.07·5-s − 4.58·7-s + 7.73·9-s − 5.64·11-s − 5.67·13-s − 3.51·15-s + 4.49·17-s + 2.37·19-s − 15.0·21-s + 6.19·23-s − 3.84·25-s + 15.5·27-s + 8.27·29-s − 3.72·31-s − 18.4·33-s + 4.92·35-s + 2.36·37-s − 18.5·39-s + 6.44·41-s + 1.50·43-s − 8.30·45-s − 2.22·47-s + 13.9·49-s + 14.7·51-s + 5.90·53-s + 6.06·55-s + ⋯ |
| L(s) = 1 | + 1.89·3-s − 0.480·5-s − 1.73·7-s + 2.57·9-s − 1.70·11-s − 1.57·13-s − 0.908·15-s + 1.08·17-s + 0.544·19-s − 3.27·21-s + 1.29·23-s − 0.769·25-s + 2.98·27-s + 1.53·29-s − 0.669·31-s − 3.21·33-s + 0.831·35-s + 0.388·37-s − 2.97·39-s + 1.00·41-s + 0.229·43-s − 1.23·45-s − 0.324·47-s + 1.99·49-s + 2.06·51-s + 0.810·53-s + 0.817·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.540937037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540937037\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 - 5.90T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 3.35T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 - 2.05T + 97T^{2} \) |
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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
−7.77127942376819029452947960742, −7.33794677630781740724257094771, −6.91108893777195842463184697592, −5.69000971215209959969522277438, −4.87337339557532810054432726932, −4.01292673385721155113924141580, −3.02408640692876227545432033961, −3.00109673459653687711868180408, −2.27573919513628334756701405668, −0.67891750404729943616230164604,
0.67891750404729943616230164604, 2.27573919513628334756701405668, 3.00109673459653687711868180408, 3.02408640692876227545432033961, 4.01292673385721155113924141580, 4.87337339557532810054432726932, 5.69000971215209959969522277438, 6.91108893777195842463184697592, 7.33794677630781740724257094771, 7.77127942376819029452947960742
