| L(s) = 1 | − 1.24·2-s − 3-s − 0.444·4-s − 3.92·5-s + 1.24·6-s − 1.90·7-s + 3.04·8-s + 9-s + 4.89·10-s + 1.97·11-s + 0.444·12-s + 3.57·13-s + 2.37·14-s + 3.92·15-s − 2.91·16-s − 17-s − 1.24·18-s − 5.19·19-s + 1.74·20-s + 1.90·21-s − 2.45·22-s − 3.01·23-s − 3.04·24-s + 10.4·25-s − 4.45·26-s − 27-s + 0.845·28-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.577·3-s − 0.222·4-s − 1.75·5-s + 0.509·6-s − 0.718·7-s + 1.07·8-s + 0.333·9-s + 1.54·10-s + 0.594·11-s + 0.128·12-s + 0.991·13-s + 0.633·14-s + 1.01·15-s − 0.728·16-s − 0.242·17-s − 0.293·18-s − 1.19·19-s + 0.389·20-s + 0.415·21-s − 0.523·22-s − 0.627·23-s − 0.622·24-s + 2.08·25-s − 0.874·26-s − 0.192·27-s + 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 1.97T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 + 2.36T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 + 0.968T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 + 5.13T + 59T^{2} \) |
| 61 | \( 1 + 0.466T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.62192585900616383634257357833, −6.88512838947099360626941752803, −6.41898119007764234637720522134, −5.38364292062196386036630189266, −4.42049262754958559609103887391, −3.92046296783485568251718742455, −3.45606430807740034524885078567, −1.88516298385803381462518745822, −0.73522570622780817045919465050, 0,
0.73522570622780817045919465050, 1.88516298385803381462518745822, 3.45606430807740034524885078567, 3.92046296783485568251718742455, 4.42049262754958559609103887391, 5.38364292062196386036630189266, 6.41898119007764234637720522134, 6.88512838947099360626941752803, 7.62192585900616383634257357833
