| L(s) = 1 | + 3-s + 3.46·5-s − 1.73·7-s + 9-s + 3.46·11-s + 3.46·15-s − 3.46·19-s − 1.73·21-s − 6·23-s + 6.99·25-s + 27-s + 6·29-s − 1.73·31-s + 3.46·33-s − 5.99·35-s + 6.92·41-s − 43-s + 3.46·45-s + 3.46·47-s − 4·49-s + 12·53-s + 11.9·55-s − 3.46·57-s − 3.46·59-s + 61-s − 1.73·63-s + 8.66·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.54·5-s − 0.654·7-s + 0.333·9-s + 1.04·11-s + 0.894·15-s − 0.794·19-s − 0.377·21-s − 1.25·23-s + 1.39·25-s + 0.192·27-s + 1.11·29-s − 0.311·31-s + 0.603·33-s − 1.01·35-s + 1.08·41-s − 0.152·43-s + 0.516·45-s + 0.505·47-s − 0.571·49-s + 1.64·53-s + 1.61·55-s − 0.458·57-s − 0.450·59-s + 0.128·61-s − 0.218·63-s + 1.05·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.601230636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.601230636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−7.914742258493993233136385732467, −6.91902854097483798828469619614, −6.35644948737587482518629829947, −6.01124298476723272419499876157, −5.08143603512056640526757292127, −4.15786349006164698609965746421, −3.48934529834465575486508023284, −2.43042029089779197788124239944, −2.00579167201248311567823960584, −0.931919197640142301796455455128,
0.931919197640142301796455455128, 2.00579167201248311567823960584, 2.43042029089779197788124239944, 3.48934529834465575486508023284, 4.15786349006164698609965746421, 5.08143603512056640526757292127, 6.01124298476723272419499876157, 6.35644948737587482518629829947, 6.91902854097483798828469619614, 7.914742258493993233136385732467
