| L(s) = 1 | − 5-s + 3.58·7-s + 1.35·13-s − 5.58·17-s + 19-s + 4.87·23-s + 25-s − 9.58·29-s − 7.17·31-s − 3.58·35-s − 0.945·37-s − 10.4·41-s − 2.71·43-s + 5.89·47-s + 5.87·49-s − 9.81·53-s − 10.1·59-s + 3.28·61-s − 1.35·65-s + 10.3·67-s − 14.3·71-s − 4.15·73-s + 1.28·79-s + 11.1·83-s + 5.58·85-s + 6.45·89-s + 4.87·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.35·7-s + 0.376·13-s − 1.35·17-s + 0.229·19-s + 1.01·23-s + 0.200·25-s − 1.78·29-s − 1.28·31-s − 0.606·35-s − 0.155·37-s − 1.63·41-s − 0.414·43-s + 0.859·47-s + 0.838·49-s − 1.34·53-s − 1.32·59-s + 0.420·61-s − 0.168·65-s + 1.26·67-s − 1.70·71-s − 0.486·73-s + 0.144·79-s + 1.22·83-s + 0.606·85-s + 0.684·89-s + 0.510·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.56304462327176562054618198892, −7.12093317200895291939827470452, −6.24291439149481753472744661947, −5.27799726626369795694244766756, −4.84137733231064422072902447162, −4.02048812623273894161116574959, −3.28491002742818767014429184688, −2.08792391132687976638181635199, −1.44364647905039796473286476321, 0,
1.44364647905039796473286476321, 2.08792391132687976638181635199, 3.28491002742818767014429184688, 4.02048812623273894161116574959, 4.84137733231064422072902447162, 5.27799726626369795694244766756, 6.24291439149481753472744661947, 7.12093317200895291939827470452, 7.56304462327176562054618198892
