| L(s) = 1 | + 0.484·3-s + 5-s + 0.484·7-s − 2.76·9-s + 5.28·13-s + 0.484·15-s + 2.48·17-s + 4.73·19-s + 0.235·21-s − 4.24·23-s + 25-s − 2.79·27-s − 3.76·29-s + 0.235·31-s + 0.484·35-s + 5.76·37-s + 2.56·39-s − 0.969·41-s + 0.249·43-s − 2.76·45-s − 3.28·47-s − 6.76·49-s + 1.20·51-s + 5.76·53-s + 2.29·57-s + 12.4·59-s + 9.70·61-s + ⋯ |
| L(s) = 1 | + 0.279·3-s + 0.447·5-s + 0.183·7-s − 0.921·9-s + 1.46·13-s + 0.125·15-s + 0.602·17-s + 1.08·19-s + 0.0513·21-s − 0.886·23-s + 0.200·25-s − 0.537·27-s − 0.699·29-s + 0.0422·31-s + 0.0819·35-s + 0.947·37-s + 0.409·39-s − 0.151·41-s + 0.0380·43-s − 0.412·45-s − 0.478·47-s − 0.966·49-s + 0.168·51-s + 0.791·53-s + 0.304·57-s + 1.62·59-s + 1.24·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.710786043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.710786043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.484T + 3T^{2} \) |
| 7 | \( 1 - 0.484T + 7T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 0.235T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 0.969T + 41T^{2} \) |
| 43 | \( 1 - 0.249T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 5.70T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.83828451093041451794879831929, −7.00436040325352084071869478024, −6.14942655120269324759598888743, −5.68843655042114507259147379968, −5.14220123066984480179456506416, −3.97163335420475411829806373435, −3.45335110407644180131352012751, −2.63556625974732090561476218092, −1.73359930492885931557187196391, −0.796401126466010236973448118470,
0.796401126466010236973448118470, 1.73359930492885931557187196391, 2.63556625974732090561476218092, 3.45335110407644180131352012751, 3.97163335420475411829806373435, 5.14220123066984480179456506416, 5.68843655042114507259147379968, 6.14942655120269324759598888743, 7.00436040325352084071869478024, 7.83828451093041451794879831929
