| L(s) = 1 | + 2·5-s − 4·7-s − 4·11-s + 4·17-s + 4·19-s + 4·23-s − 25-s − 2·29-s + 4·31-s − 8·35-s + 12·37-s − 12·41-s + 8·43-s + 9·49-s + 14·53-s − 8·55-s + 2·59-s + 2·61-s + 4·67-s − 8·71-s + 6·73-s + 16·77-s − 14·79-s + 6·83-s + 8·85-s + 6·89-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.20·11-s + 0.970·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 1.35·35-s + 1.97·37-s − 1.87·41-s + 1.21·43-s + 9/7·49-s + 1.92·53-s − 1.07·55-s + 0.260·59-s + 0.256·61-s + 0.488·67-s − 0.949·71-s + 0.702·73-s + 1.82·77-s − 1.57·79-s + 0.658·83-s + 0.867·85-s + 0.635·89-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.353661915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.353661915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.47421158569414, −13.26277762066950, −13.15474909859936, −12.32801760498628, −12.00895316571790, −11.32959988510383, −10.67906786144607, −10.14031702726534, −9.849354150837251, −9.557329959127949, −8.942391934527236, −8.315937413418018, −7.664953091135551, −7.226504797969287, −6.689972134398567, −5.977966041442014, −5.711332811305020, −5.246818702905199, −4.540399133259975, −3.686454897541879, −3.163006411591632, −2.686210541584453, −2.179681655599040, −1.125467497111443, −0.5329564594353031,
0.5329564594353031, 1.125467497111443, 2.179681655599040, 2.686210541584453, 3.163006411591632, 3.686454897541879, 4.540399133259975, 5.246818702905199, 5.711332811305020, 5.977966041442014, 6.689972134398567, 7.226504797969287, 7.664953091135551, 8.315937413418018, 8.942391934527236, 9.557329959127949, 9.849354150837251, 10.14031702726534, 10.67906786144607, 11.32959988510383, 12.00895316571790, 12.32801760498628, 13.15474909859936, 13.26277762066950, 13.47421158569414
