| L(s) = 1 | − 2.76·2-s + 2.33·3-s + 5.62·4-s − 2.36·5-s − 6.45·6-s + 1.79·7-s − 10.0·8-s + 2.46·9-s + 6.53·10-s − 3.92·11-s + 13.1·12-s + 7.17·13-s − 4.95·14-s − 5.52·15-s + 16.3·16-s − 5.22·17-s − 6.79·18-s + 5.03·19-s − 13.3·20-s + 4.19·21-s + 10.8·22-s + 2.81·23-s − 23.3·24-s + 0.593·25-s − 19.8·26-s − 1.25·27-s + 10.0·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 1.34·3-s + 2.81·4-s − 1.05·5-s − 2.63·6-s + 0.678·7-s − 3.53·8-s + 0.820·9-s + 2.06·10-s − 1.18·11-s + 3.79·12-s + 1.98·13-s − 1.32·14-s − 1.42·15-s + 4.09·16-s − 1.26·17-s − 1.60·18-s + 1.15·19-s − 2.97·20-s + 0.915·21-s + 2.31·22-s + 0.586·23-s − 4.77·24-s + 0.118·25-s − 3.88·26-s − 0.242·27-s + 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.118097582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118097582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 13 | \( 1 - 7.17T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 - 9.20T + 47T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 - 5.58T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.093077365243690436324135584795, −7.55449350208338859150133202353, −7.04157624345442425054478450077, −6.14774870050260069551942809669, −5.10292896963823579113417855485, −3.74717010246107785228606918836, −3.26973776745459324753891215508, −2.41830667282396001487608112175, −1.66251942098271053761957859557, −0.66189051859186061178363049554,
0.66189051859186061178363049554, 1.66251942098271053761957859557, 2.41830667282396001487608112175, 3.26973776745459324753891215508, 3.74717010246107785228606918836, 5.10292896963823579113417855485, 6.14774870050260069551942809669, 7.04157624345442425054478450077, 7.55449350208338859150133202353, 8.093077365243690436324135584795
