| L(s) = 1 | + 3.35·3-s − 5-s − 3.81·7-s + 8.24·9-s + 0.459·13-s − 3.35·15-s − 7.78·17-s + 1.54·19-s − 12.7·21-s − 8.24·23-s + 25-s + 17.6·27-s − 6.70·29-s − 7.54·31-s + 3.81·35-s + 4.70·37-s + 1.54·39-s − 1.45·41-s + 3.97·43-s − 8.24·45-s + 0.272·47-s + 7.54·49-s − 26.1·51-s − 2.62·53-s + 5.16·57-s − 8.24·59-s + 0.751·61-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 0.447·5-s − 1.44·7-s + 2.74·9-s + 0.127·13-s − 0.865·15-s − 1.88·17-s + 0.353·19-s − 2.79·21-s − 1.71·23-s + 0.200·25-s + 3.38·27-s − 1.24·29-s − 1.35·31-s + 0.644·35-s + 0.773·37-s + 0.246·39-s − 0.227·41-s + 0.606·43-s − 1.22·45-s + 0.0397·47-s + 1.07·49-s − 3.65·51-s − 0.360·53-s + 0.684·57-s − 1.07·59-s + 0.0962·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 13 | \( 1 - 0.459T + 13T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 - 0.272T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 0.751T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 0.918T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.459T + 97T^{2} \) |
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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
−7.901898471053969567117110480427, −7.38766459669942561281154620972, −6.71141433712892672640809671801, −5.95148742667249809249055349404, −4.47673172466023877623638578164, −3.90131488884499232120354911703, −3.34986226032756410216320173333, −2.51813604807878510736711395143, −1.80376588206215101272348193451, 0,
1.80376588206215101272348193451, 2.51813604807878510736711395143, 3.34986226032756410216320173333, 3.90131488884499232120354911703, 4.47673172466023877623638578164, 5.95148742667249809249055349404, 6.71141433712892672640809671801, 7.38766459669942561281154620972, 7.901898471053969567117110480427
