| L(s) = 1 | − 2-s + 0.803·3-s + 4-s + 5-s − 0.803·6-s − 2.71·7-s − 8-s − 2.35·9-s − 10-s + 3.99·11-s + 0.803·12-s − 13-s + 2.71·14-s + 0.803·15-s + 16-s + 5.09·17-s + 2.35·18-s − 5.23·19-s + 20-s − 2.17·21-s − 3.99·22-s + 0.963·23-s − 0.803·24-s + 25-s + 26-s − 4.30·27-s − 2.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.463·3-s + 0.5·4-s + 0.447·5-s − 0.328·6-s − 1.02·7-s − 0.353·8-s − 0.784·9-s − 0.316·10-s + 1.20·11-s + 0.231·12-s − 0.277·13-s + 0.724·14-s + 0.207·15-s + 0.250·16-s + 1.23·17-s + 0.554·18-s − 1.20·19-s + 0.223·20-s − 0.475·21-s − 0.852·22-s + 0.200·23-s − 0.164·24-s + 0.200·25-s + 0.196·26-s − 0.828·27-s − 0.512·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.803T + 3T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 0.963T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 37 | \( 1 + 8.00T + 37T^{2} \) |
| 41 | \( 1 - 0.805T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 - 2.27T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 - 0.107T + 67T^{2} \) |
| 71 | \( 1 + 7.64T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 - 0.933T + 79T^{2} \) |
| 83 | \( 1 + 3.88T + 83T^{2} \) |
| 89 | \( 1 - 0.741T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.351589069793324603139163064863, −7.33435684939168669640245767927, −6.68009408006341008980422622669, −6.04674411533945086061514706574, −5.33507333999098889484765551689, −3.93529020277556553064454538753, −3.27152164542985850410035024549, −2.43599377056447763332239221040, −1.41410793786810822290092195585, 0,
1.41410793786810822290092195585, 2.43599377056447763332239221040, 3.27152164542985850410035024549, 3.93529020277556553064454538753, 5.33507333999098889484765551689, 6.04674411533945086061514706574, 6.68009408006341008980422622669, 7.33435684939168669640245767927, 8.351589069793324603139163064863
