L(s) = 1 | + 2-s + 0.885·3-s + 4-s − 1.00·5-s + 0.885·6-s + 2.56·7-s + 8-s − 2.21·9-s − 1.00·10-s + 2.77·11-s + 0.885·12-s + 5.85·13-s + 2.56·14-s − 0.885·15-s + 16-s + 2.27·17-s − 2.21·18-s − 0.846·19-s − 1.00·20-s + 2.26·21-s + 2.77·22-s + 0.431·23-s + 0.885·24-s − 3.99·25-s + 5.85·26-s − 4.61·27-s + 2.56·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.511·3-s + 0.5·4-s − 0.447·5-s + 0.361·6-s + 0.967·7-s + 0.353·8-s − 0.738·9-s − 0.316·10-s + 0.837·11-s + 0.255·12-s + 1.62·13-s + 0.684·14-s − 0.228·15-s + 0.250·16-s + 0.551·17-s − 0.522·18-s − 0.194·19-s − 0.223·20-s + 0.494·21-s + 0.592·22-s + 0.0899·23-s + 0.180·24-s − 0.799·25-s + 1.14·26-s − 0.888·27-s + 0.483·28-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
\(L(1)\) |
\(\approx\) |
\(4.154718092\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.154718092\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.885T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 + 0.846T + 19T^{2} \) |
| 23 | \( 1 - 0.431T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 0.383T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 - 8.19T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.299329424318420898461358031934, −7.962207029694079966706731006678, −6.87360410625374846676465655240, −6.16316023857386415474634602139, −5.45736580895130526989458054384, −4.54261043230882256311185539920, −3.75990987559138110569625588540, −3.24370280781482558890928196045, −2.06887251272903720811522971195, −1.11865637425401466323895845354,
1.11865637425401466323895845354, 2.06887251272903720811522971195, 3.24370280781482558890928196045, 3.75990987559138110569625588540, 4.54261043230882256311185539920, 5.45736580895130526989458054384, 6.16316023857386415474634602139, 6.87360410625374846676465655240, 7.962207029694079966706731006678, 8.299329424318420898461358031934