L(s) = 1 | − 2-s − 2.77·3-s + 4-s − 2.42·5-s + 2.77·6-s − 4.56·7-s − 8-s + 4.72·9-s + 2.42·10-s + 4.27·11-s − 2.77·12-s + 4.31·13-s + 4.56·14-s + 6.73·15-s + 16-s − 2.17·17-s − 4.72·18-s − 0.518·19-s − 2.42·20-s + 12.6·21-s − 4.27·22-s + 5.19·23-s + 2.77·24-s + 0.879·25-s − 4.31·26-s − 4.79·27-s − 4.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.60·3-s + 0.5·4-s − 1.08·5-s + 1.13·6-s − 1.72·7-s − 0.353·8-s + 1.57·9-s + 0.766·10-s + 1.28·11-s − 0.802·12-s + 1.19·13-s + 1.21·14-s + 1.73·15-s + 0.250·16-s − 0.528·17-s − 1.11·18-s − 0.118·19-s − 0.542·20-s + 2.76·21-s − 0.911·22-s + 1.08·23-s + 0.567·24-s + 0.175·25-s − 0.845·26-s − 0.921·27-s − 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2813729755\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2813729755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 0.518T + 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 + 7.54T + 29T^{2} \) |
| 31 | \( 1 - 2.72T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 - 0.808T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.618490571774471435526763324114, −7.44394999237269182947981279686, −6.88624624085683852717030477972, −6.20793545627270492102785767683, −6.00152979459993553915005415295, −4.63790769052242382706539037035, −3.81850358858607412530969403552, −3.18552771650538170487224289859, −1.37973964287166614706043975908, −0.39925024409845837390399578857,
0.39925024409845837390399578857, 1.37973964287166614706043975908, 3.18552771650538170487224289859, 3.81850358858607412530969403552, 4.63790769052242382706539037035, 6.00152979459993553915005415295, 6.20793545627270492102785767683, 6.88624624085683852717030477972, 7.44394999237269182947981279686, 8.618490571774471435526763324114