| L(s) = 1 | − 1.96·3-s − 3.06·5-s − 7-s + 0.871·9-s + 4.37·11-s − 2.48·13-s + 6.02·15-s + 5.20·17-s − 1.20·19-s + 1.96·21-s − 6.15·23-s + 4.37·25-s + 4.18·27-s − 8.82·29-s + 2.78·31-s − 8.61·33-s + 3.06·35-s − 5.81·37-s + 4.88·39-s + 6.32·41-s − 4.32·43-s − 2.66·45-s + 3.60·47-s + 49-s − 10.2·51-s + 7.47·53-s − 13.4·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s − 1.36·5-s − 0.377·7-s + 0.290·9-s + 1.31·11-s − 0.688·13-s + 1.55·15-s + 1.26·17-s − 0.276·19-s + 0.429·21-s − 1.28·23-s + 0.874·25-s + 0.806·27-s − 1.63·29-s + 0.499·31-s − 1.49·33-s + 0.517·35-s − 0.956·37-s + 0.781·39-s + 0.988·41-s − 0.659·43-s − 0.397·45-s + 0.525·47-s + 0.142·49-s − 1.43·51-s + 1.02·53-s − 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4464477218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4464477218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 1.36T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 3.26T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−7.78737505046526866615632074469, −7.18400607500829682507688433584, −6.54131771199626141386046610356, −5.79861249005910194505609710126, −5.23030254242118813464884300283, −4.11023356176599280727665680521, −3.92150796031766589702720003636, −2.90049174520043503149188350344, −1.49310773258850631740108781542, −0.37306180084597189159246323325,
0.37306180084597189159246323325, 1.49310773258850631740108781542, 2.90049174520043503149188350344, 3.92150796031766589702720003636, 4.11023356176599280727665680521, 5.23030254242118813464884300283, 5.79861249005910194505609710126, 6.54131771199626141386046610356, 7.18400607500829682507688433584, 7.78737505046526866615632074469
