L(s) = 1 | − 3.01·3-s + 3.98·5-s + 4.01·7-s + 6.07·9-s − 1.62·11-s + 0.371·13-s − 12.0·15-s + 4.12·17-s + 6.23·19-s − 12.1·21-s − 1.25·23-s + 10.9·25-s − 9.25·27-s + 4.56·29-s − 3.83·31-s + 4.89·33-s + 16.0·35-s + 5.64·37-s − 1.11·39-s − 8.40·41-s + 0.628·43-s + 24.2·45-s + 10.0·47-s + 9.14·49-s − 12.4·51-s + 7.64·53-s − 6.48·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.51·7-s + 2.02·9-s − 0.490·11-s + 0.102·13-s − 3.10·15-s + 1.00·17-s + 1.42·19-s − 2.64·21-s − 0.260·23-s + 2.18·25-s − 1.78·27-s + 0.847·29-s − 0.688·31-s + 0.852·33-s + 2.70·35-s + 0.928·37-s − 0.178·39-s − 1.31·41-s + 0.0958·43-s + 3.60·45-s + 1.46·47-s + 1.30·49-s − 1.74·51-s + 1.04·53-s − 0.874·55-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
\(L(1)\) |
\(\approx\) |
\(2.095944703\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.095944703\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 0.371T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 - 0.628T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 2.15T + 83T^{2} \) |
| 89 | \( 1 - 8.00T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.416125101765493409482568863931, −7.48785892119988203065438771311, −6.84470383007059697801257833912, −5.84739327455916319132084794240, −5.45236849147856397947755813050, −5.18940237364768255519031591046, −4.27414172174715154584473574152, −2.68208390529119643345358392003, −1.54612308837083997841025302563, −1.05611100049065067351691676016,
1.05611100049065067351691676016, 1.54612308837083997841025302563, 2.68208390529119643345358392003, 4.27414172174715154584473574152, 5.18940237364768255519031591046, 5.45236849147856397947755813050, 5.84739327455916319132084794240, 6.84470383007059697801257833912, 7.48785892119988203065438771311, 8.416125101765493409482568863931