| L(s) = 1 | + 2-s − 1.93·3-s + 4-s − 1.16·5-s − 1.93·6-s + 8-s + 0.730·9-s − 1.16·10-s + 1.40·11-s − 1.93·12-s − 4.56·13-s + 2.25·15-s + 16-s + 4.59·17-s + 0.730·18-s − 1.15·19-s − 1.16·20-s + 1.40·22-s + 6.00·23-s − 1.93·24-s − 3.63·25-s − 4.56·26-s + 4.38·27-s + 4.33·29-s + 2.25·30-s − 7.01·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s − 0.521·5-s − 0.788·6-s + 0.353·8-s + 0.243·9-s − 0.368·10-s + 0.424·11-s − 0.557·12-s − 1.26·13-s + 0.581·15-s + 0.250·16-s + 1.11·17-s + 0.172·18-s − 0.265·19-s − 0.260·20-s + 0.300·22-s + 1.25·23-s − 0.394·24-s − 0.727·25-s − 0.895·26-s + 0.843·27-s + 0.804·29-s + 0.411·30-s − 1.25·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 + 7.01T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 - 7.27T + 47T^{2} \) |
| 53 | \( 1 + 0.473T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 7.11T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + 3.31T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 5.29T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 9.35T + 97T^{2} \) |
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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
−7.67474106430436489930009592830, −7.31783657596235887585104096664, −6.40275398754683023066341042683, −5.79671790472492749869634329440, −5.02159562649917379022101793534, −4.54445003380316178752522364947, −3.51383376365401997987970630270, −2.68555546137192548016884960269, −1.31164912154202001776332293451, 0,
1.31164912154202001776332293451, 2.68555546137192548016884960269, 3.51383376365401997987970630270, 4.54445003380316178752522364947, 5.02159562649917379022101793534, 5.79671790472492749869634329440, 6.40275398754683023066341042683, 7.31783657596235887585104096664, 7.67474106430436489930009592830
