| L(s) = 1 | + 2-s + 0.0521·3-s + 4-s − 1.62·5-s + 0.0521·6-s − 1.39·7-s + 8-s − 2.99·9-s − 1.62·10-s − 6.53·11-s + 0.0521·12-s − 0.259·13-s − 1.39·14-s − 0.0846·15-s + 16-s − 2.65·17-s − 2.99·18-s + 7.28·19-s − 1.62·20-s − 0.0727·21-s − 6.53·22-s + 2.12·23-s + 0.0521·24-s − 2.36·25-s − 0.259·26-s − 0.312·27-s − 1.39·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0301·3-s + 0.5·4-s − 0.725·5-s + 0.0212·6-s − 0.526·7-s + 0.353·8-s − 0.999·9-s − 0.513·10-s − 1.97·11-s + 0.0150·12-s − 0.0720·13-s − 0.372·14-s − 0.0218·15-s + 0.250·16-s − 0.644·17-s − 0.706·18-s + 1.67·19-s − 0.362·20-s − 0.0158·21-s − 1.39·22-s + 0.443·23-s + 0.0106·24-s − 0.472·25-s − 0.0509·26-s − 0.0602·27-s − 0.263·28-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
| \(L(1)\) |
\(\approx\) |
\(1.654735188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.654735188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.0521T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 + 0.259T + 13T^{2} \) |
| 17 | \( 1 + 2.65T + 17T^{2} \) |
| 19 | \( 1 - 7.28T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 0.536T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.224207744446337574957396294834, −7.68102922237464050038321942604, −7.05582775628578709783583889530, −6.02797168879542018369389446620, −5.40994071182226417796311188870, −4.78414859783301829576852788937, −3.78889503567282539629928949479, −2.83211055852391315926455406541, −2.58711191821272505794478823397, −0.62442784880626553382881400126,
0.62442784880626553382881400126, 2.58711191821272505794478823397, 2.83211055852391315926455406541, 3.78889503567282539629928949479, 4.78414859783301829576852788937, 5.40994071182226417796311188870, 6.02797168879542018369389446620, 7.05582775628578709783583889530, 7.68102922237464050038321942604, 8.224207744446337574957396294834
