| L(s) = 1 | + 3·3-s − 5·5-s − 32·7-s + 9·9-s + 64·11-s + 6·13-s − 15·15-s + 38·17-s − 116·19-s − 96·21-s + 120·23-s + 25·25-s + 27·27-s + 122·29-s − 164·31-s + 192·33-s + 160·35-s − 146·37-s + 18·39-s − 238·41-s − 148·43-s − 45·45-s + 184·47-s + 681·49-s + 114·51-s − 470·53-s − 320·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.72·7-s + 1/3·9-s + 1.75·11-s + 0.128·13-s − 0.258·15-s + 0.542·17-s − 1.40·19-s − 0.997·21-s + 1.08·23-s + 1/5·25-s + 0.192·27-s + 0.781·29-s − 0.950·31-s + 1.01·33-s + 0.772·35-s − 0.648·37-s + 0.0739·39-s − 0.906·41-s − 0.524·43-s − 0.149·45-s + 0.571·47-s + 1.98·49-s + 0.313·51-s − 1.21·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 64 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 238 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 470 T + p^{3} T^{2} \) |
| 59 | \( 1 + 216 T + p^{3} T^{2} \) |
| 61 | \( 1 + 806 T + p^{3} T^{2} \) |
| 67 | \( 1 + 732 T + p^{3} T^{2} \) |
| 71 | \( 1 + 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 638 T + p^{3} T^{2} \) |
| 79 | \( 1 + 596 T + p^{3} T^{2} \) |
| 83 | \( 1 + 884 T + p^{3} T^{2} \) |
| 89 | \( 1 - 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 322 T + p^{3} T^{2} \) |
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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
−8.994766648039064603402155431079, −8.797383311100631773806072546619, −7.39236389129581426299580198067, −6.66439332125314721179710650272, −6.10412010793020609218993618592, −4.50058561833425545711831580768, −3.60406893502746389783375761063, −3.02477919016310366956666027081, −1.45176078444251877423939438451, 0,
1.45176078444251877423939438451, 3.02477919016310366956666027081, 3.60406893502746389783375761063, 4.50058561833425545711831580768, 6.10412010793020609218993618592, 6.66439332125314721179710650272, 7.39236389129581426299580198067, 8.797383311100631773806072546619, 8.994766648039064603402155431079
