| L(s) = 1 | − 2-s − 3-s + 4-s + 3.26·5-s + 6-s − 3.10·7-s − 8-s + 9-s − 3.26·10-s − 4.62·11-s − 12-s + 0.359·13-s + 3.10·14-s − 3.26·15-s + 16-s + 3.61·17-s − 18-s − 3.61·19-s + 3.26·20-s + 3.10·21-s + 4.62·22-s − 23-s + 24-s + 5.67·25-s − 0.359·26-s − 27-s − 3.10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s + 0.408·6-s − 1.17·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s − 1.39·11-s − 0.288·12-s + 0.0996·13-s + 0.830·14-s − 0.843·15-s + 0.250·16-s + 0.876·17-s − 0.235·18-s − 0.828·19-s + 0.730·20-s + 0.677·21-s + 0.986·22-s − 0.208·23-s + 0.204·24-s + 1.13·25-s − 0.0704·26-s − 0.192·27-s − 0.586·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 4.62T + 11T^{2} \) |
| 13 | \( 1 - 0.359T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 31 | \( 1 - 7.25T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 - 1.96T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 6.85T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.78T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 1.92T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.074525500270647006879947119253, −7.35262677758115835408673070339, −6.39859029085102561557047152657, −6.00688594632224352907434517703, −5.47262664438804258936015800793, −4.39927174011558459933891701962, −2.97275004730072728481209975362, −2.46244279937793055891075075847, −1.27576154323326769908535480658, 0,
1.27576154323326769908535480658, 2.46244279937793055891075075847, 2.97275004730072728481209975362, 4.39927174011558459933891701962, 5.47262664438804258936015800793, 6.00688594632224352907434517703, 6.39859029085102561557047152657, 7.35262677758115835408673070339, 8.074525500270647006879947119253
