| L(s) = 1 | − 1.12·3-s − 3.66·5-s + 0.118·7-s − 1.72·9-s + 0.229·11-s + 2.85·13-s + 4.13·15-s + 4.90·17-s − 19-s − 0.133·21-s − 2.13·23-s + 8.44·25-s + 5.33·27-s − 7.59·29-s − 1.48·31-s − 0.258·33-s − 0.434·35-s − 11.3·37-s − 3.21·39-s + 7.15·41-s − 10.4·43-s + 6.33·45-s + 3.54·47-s − 6.98·49-s − 5.54·51-s − 0.184·53-s − 0.841·55-s + ⋯ |
| L(s) = 1 | − 0.651·3-s − 1.63·5-s + 0.0448·7-s − 0.575·9-s + 0.0691·11-s + 0.790·13-s + 1.06·15-s + 1.19·17-s − 0.229·19-s − 0.0291·21-s − 0.444·23-s + 1.68·25-s + 1.02·27-s − 1.41·29-s − 0.266·31-s − 0.0450·33-s − 0.0734·35-s − 1.87·37-s − 0.515·39-s + 1.11·41-s − 1.60·43-s + 0.943·45-s + 0.517·47-s − 0.997·49-s − 0.775·51-s − 0.0254·53-s − 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5646447905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5646447905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 7 | \( 1 - 0.118T + 7T^{2} \) |
| 11 | \( 1 - 0.229T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 4.90T + 17T^{2} \) |
| 23 | \( 1 + 2.13T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 7.15T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.54T + 47T^{2} \) |
| 53 | \( 1 + 0.184T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 6.54T + 67T^{2} \) |
| 71 | \( 1 - 0.940T + 71T^{2} \) |
| 73 | \( 1 - 8.23T + 73T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 2.19T + 97T^{2} \) |
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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
−8.011618554052847464119670460706, −7.48427826908155938919360251523, −6.68895572985037685772325612995, −5.88814441780178486342887884475, −5.25116267791123932346136424791, −4.41386528359270486106129678813, −3.56008078534865636601050317225, −3.19283117364467735286612234817, −1.62887678352852458710628934618, −0.41264664132178376217951346900,
0.41264664132178376217951346900, 1.62887678352852458710628934618, 3.19283117364467735286612234817, 3.56008078534865636601050317225, 4.41386528359270486106129678813, 5.25116267791123932346136424791, 5.88814441780178486342887884475, 6.68895572985037685772325612995, 7.48427826908155938919360251523, 8.011618554052847464119670460706
