| L(s) = 1 | + 1.53·3-s + 5-s + 1.53·7-s − 0.633·9-s + 4.57·11-s − 3.63·13-s + 1.53·15-s + 1.77·17-s + 5.17·19-s + 2.36·21-s + 4.20·23-s + 25-s − 5.58·27-s − 29-s + 1.17·31-s + 7.03·33-s + 1.53·35-s − 3.36·37-s − 5.58·39-s − 1.95·41-s + 2.46·43-s − 0.633·45-s + 7.12·47-s − 4.63·49-s + 2.73·51-s − 8.11·53-s + 4.57·55-s + ⋯ |
| L(s) = 1 | + 0.888·3-s + 0.447·5-s + 0.581·7-s − 0.211·9-s + 1.37·11-s − 1.00·13-s + 0.397·15-s + 0.430·17-s + 1.18·19-s + 0.516·21-s + 0.876·23-s + 0.200·25-s − 1.07·27-s − 0.185·29-s + 0.210·31-s + 1.22·33-s + 0.260·35-s − 0.553·37-s − 0.895·39-s − 0.305·41-s + 0.375·43-s − 0.0943·45-s + 1.03·47-s − 0.661·49-s + 0.382·51-s − 1.11·53-s + 0.616·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.807830298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.807830298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.36T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 8.11T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 8.77T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.97T + 79T^{2} \) |
| 83 | \( 1 + 3.60T + 83T^{2} \) |
| 89 | \( 1 - 9.74T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.81043774106641902958999159996, −7.07855440676444547941141498801, −6.52111974763174388173411458647, −5.44286119266726511944148010147, −5.10485675168862687461627363674, −4.05649697773142171756300405093, −3.37521393701011575178590349768, −2.62413560403365195308467863999, −1.82749839305297048500834232014, −0.940564823738262941429470648808,
0.940564823738262941429470648808, 1.82749839305297048500834232014, 2.62413560403365195308467863999, 3.37521393701011575178590349768, 4.05649697773142171756300405093, 5.10485675168862687461627363674, 5.44286119266726511944148010147, 6.52111974763174388173411458647, 7.07855440676444547941141498801, 7.81043774106641902958999159996
