| L(s) = 1 | − 2.70·2-s + 3.13·3-s + 5.29·4-s − 0.910·5-s − 8.45·6-s + 0.335·7-s − 8.89·8-s + 6.80·9-s + 2.45·10-s − 2.87·11-s + 16.5·12-s + 1.35·13-s − 0.904·14-s − 2.84·15-s + 13.4·16-s − 1.80·17-s − 18.3·18-s + 6.84·19-s − 4.81·20-s + 1.04·21-s + 7.75·22-s − 6.94·23-s − 27.8·24-s − 4.17·25-s − 3.66·26-s + 11.8·27-s + 1.77·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 1.80·3-s + 2.64·4-s − 0.407·5-s − 3.45·6-s + 0.126·7-s − 3.14·8-s + 2.26·9-s + 0.777·10-s − 0.865·11-s + 4.78·12-s + 0.376·13-s − 0.241·14-s − 0.735·15-s + 3.35·16-s − 0.437·17-s − 4.32·18-s + 1.57·19-s − 1.07·20-s + 0.228·21-s + 1.65·22-s − 1.44·23-s − 5.68·24-s − 0.834·25-s − 0.719·26-s + 2.28·27-s + 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.910T + 5T^{2} \) |
| 7 | \( 1 - 0.335T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 + 7.95T + 29T^{2} \) |
| 31 | \( 1 - 0.990T + 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 67 | \( 1 - 1.37T + 67T^{2} \) |
| 71 | \( 1 - 2.67T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.267745811118015512295633073613, −7.70671792233160646958523142967, −7.25847904701733983069688351664, −6.39599654336509506139044982437, −5.13483977822433062723489985962, −3.55681381142236321018729126362, −3.24814562266112609695399998043, −2.03640964664739930751568461611, −1.67604688265767133588584249777, 0,
1.67604688265767133588584249777, 2.03640964664739930751568461611, 3.24814562266112609695399998043, 3.55681381142236321018729126362, 5.13483977822433062723489985962, 6.39599654336509506139044982437, 7.25847904701733983069688351664, 7.70671792233160646958523142967, 8.267745811118015512295633073613
