| L(s) = 1 | + 0.0722·3-s + 5-s − 0.709·7-s − 2.99·9-s − 0.425·11-s + 6.40·13-s + 0.0722·15-s − 4.34·17-s + 2.76·19-s − 0.0512·21-s − 2.37·23-s + 25-s − 0.432·27-s − 1.12·29-s − 1.10·31-s − 0.0307·33-s − 0.709·35-s − 7.75·37-s + 0.462·39-s − 4.05·41-s − 9.49·43-s − 2.99·45-s + 4.25·47-s − 6.49·49-s − 0.313·51-s − 5.36·53-s − 0.425·55-s + ⋯ |
| L(s) = 1 | + 0.0416·3-s + 0.447·5-s − 0.268·7-s − 0.998·9-s − 0.128·11-s + 1.77·13-s + 0.0186·15-s − 1.05·17-s + 0.634·19-s − 0.0111·21-s − 0.495·23-s + 0.200·25-s − 0.0833·27-s − 0.208·29-s − 0.198·31-s − 0.00534·33-s − 0.119·35-s − 1.27·37-s + 0.0741·39-s − 0.633·41-s − 1.44·43-s − 0.446·45-s + 0.620·47-s − 0.928·49-s − 0.0439·51-s − 0.736·53-s − 0.0573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0722T + 3T^{2} \) |
| 7 | \( 1 + 0.709T + 7T^{2} \) |
| 11 | \( 1 + 0.425T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 - 4.25T + 47T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 3.42T + 83T^{2} \) |
| 89 | \( 1 - 1.05T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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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
−7.917381661594527245479536513847, −6.70698528330724652379170446150, −6.42194560635951173659211795147, −5.57591344190978761678204251701, −5.03431222464838338352996406906, −3.81431608957831339685168948543, −3.32116257878130184137572840202, −2.31549546203952146857035629788, −1.39130717123968754416963849660, 0,
1.39130717123968754416963849660, 2.31549546203952146857035629788, 3.32116257878130184137572840202, 3.81431608957831339685168948543, 5.03431222464838338352996406906, 5.57591344190978761678204251701, 6.42194560635951173659211795147, 6.70698528330724652379170446150, 7.917381661594527245479536513847
