| L(s) = 1 | + 0.166·3-s + 0.731·5-s − 2.80·7-s − 2.97·9-s − 0.681·11-s + 3.25·13-s + 0.121·15-s + 2.38·17-s + 1.40·19-s − 0.465·21-s + 3.92·23-s − 4.46·25-s − 0.991·27-s − 5.36·29-s + 9.39·31-s − 0.113·33-s − 2.04·35-s + 6.25·37-s + 0.540·39-s − 6.09·41-s − 13.0·43-s − 2.17·45-s − 2.25·47-s + 0.846·49-s + 0.395·51-s + 2.79·53-s − 0.498·55-s + ⋯ |
| L(s) = 1 | + 0.0958·3-s + 0.326·5-s − 1.05·7-s − 0.990·9-s − 0.205·11-s + 0.902·13-s + 0.0313·15-s + 0.578·17-s + 0.321·19-s − 0.101·21-s + 0.818·23-s − 0.893·25-s − 0.190·27-s − 0.996·29-s + 1.68·31-s − 0.0197·33-s − 0.346·35-s + 1.02·37-s + 0.0864·39-s − 0.951·41-s − 1.98·43-s − 0.323·45-s − 0.328·47-s + 0.120·49-s + 0.0554·51-s + 0.383·53-s − 0.0672·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.166T + 3T^{2} \) |
| 5 | \( 1 - 0.731T + 5T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 + 0.681T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 1.40T + 19T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 31 | \( 1 - 9.39T + 31T^{2} \) |
| 37 | \( 1 - 6.25T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 13.0T + 43T^{2} \) |
| 47 | \( 1 + 2.25T + 47T^{2} \) |
| 53 | \( 1 - 2.79T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 + 3.24T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−7.971047516353796738340791915897, −6.78703736633652614183737733653, −6.34514116158285628815258687068, −5.64651015378151477377811810873, −5.02139942865825758971260076314, −3.78167429735360206325313370693, −3.22708778776672911711424761396, −2.51654768689134114679343403945, −1.27806985894444461984817705139, 0,
1.27806985894444461984817705139, 2.51654768689134114679343403945, 3.22708778776672911711424761396, 3.78167429735360206325313370693, 5.02139942865825758971260076314, 5.64651015378151477377811810873, 6.34514116158285628815258687068, 6.78703736633652614183737733653, 7.971047516353796738340791915897
