| L(s) = 1 | + 1.79·3-s − 5-s + 0.208·7-s + 0.208·9-s + 0.791·11-s − 13-s − 1.79·15-s + 17-s − 4.79·19-s + 0.373·21-s − 7.58·23-s − 4·25-s − 5.00·27-s + 9·29-s + 0.582·31-s + 1.41·33-s − 0.208·35-s + 0.582·37-s − 1.79·39-s − 9·43-s − 0.208·45-s − 3.58·47-s − 6.95·49-s + 1.79·51-s − 7.79·53-s − 0.791·55-s − 8.58·57-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 0.447·5-s + 0.0788·7-s + 0.0695·9-s + 0.238·11-s − 0.277·13-s − 0.462·15-s + 0.242·17-s − 1.09·19-s + 0.0815·21-s − 1.58·23-s − 0.800·25-s − 0.962·27-s + 1.67·29-s + 0.104·31-s + 0.246·33-s − 0.0352·35-s + 0.0957·37-s − 0.286·39-s − 1.37·43-s − 0.0311·45-s − 0.522·47-s − 0.993·49-s + 0.250·51-s − 1.07·53-s − 0.106·55-s − 1.13·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 0.208T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 19 | \( 1 + 4.79T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 - 0.582T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 4.41T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.103052231480263499842354308172, −7.892985702703196700993845093024, −6.71030374719761946720104240538, −6.12875766149546178316463851071, −4.99482850605282152392250394470, −4.13671124984514922475686817104, −3.47636578903107047760666768714, −2.55900356052614806103740122534, −1.72196835284784521258000404542, 0,
1.72196835284784521258000404542, 2.55900356052614806103740122534, 3.47636578903107047760666768714, 4.13671124984514922475686817104, 4.99482850605282152392250394470, 6.12875766149546178316463851071, 6.71030374719761946720104240538, 7.892985702703196700993845093024, 8.103052231480263499842354308172
