| L(s) = 1 | + 2-s + 0.732·3-s + 4-s + 5-s + 0.732·6-s + 8-s − 2.46·9-s + 10-s + 11-s + 0.732·12-s − 5.46·13-s + 0.732·15-s + 16-s − 3.46·17-s − 2.46·18-s − 3.26·19-s + 20-s + 22-s + 2.19·23-s + 0.732·24-s + 25-s − 5.46·26-s − 4·27-s − 1.26·29-s + 0.732·30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s + 0.447·5-s + 0.298·6-s + 0.353·8-s − 0.821·9-s + 0.316·10-s + 0.301·11-s + 0.211·12-s − 1.51·13-s + 0.189·15-s + 0.250·16-s − 0.840·17-s − 0.580·18-s − 0.749·19-s + 0.223·20-s + 0.213·22-s + 0.457·23-s + 0.149·24-s + 0.200·25-s − 1.07·26-s − 0.769·27-s − 0.235·29-s + 0.133·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 + 1.80T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.70912271553710485902607181766, −6.99808992915843149371867466579, −6.36521707563831699294652621241, −5.54887925905313055708180769170, −4.90734416091227199539881107346, −4.16474135063840074468076579871, −3.16314241797597160510812478529, −2.48847642666564286414010658725, −1.78919901355224655338223013203, 0,
1.78919901355224655338223013203, 2.48847642666564286414010658725, 3.16314241797597160510812478529, 4.16474135063840074468076579871, 4.90734416091227199539881107346, 5.54887925905313055708180769170, 6.36521707563831699294652621241, 6.99808992915843149371867466579, 7.70912271553710485902607181766
