| L(s) = 1 | + 1.17·2-s − 3-s − 0.615·4-s + 2.85·5-s − 1.17·6-s + 3.62·7-s − 3.07·8-s + 9-s + 3.35·10-s + 1.39·11-s + 0.615·12-s + 2.03·13-s + 4.27·14-s − 2.85·15-s − 2.38·16-s − 0.256·17-s + 1.17·18-s + 1.59·19-s − 1.75·20-s − 3.62·21-s + 1.64·22-s + 23-s + 3.07·24-s + 3.14·25-s + 2.39·26-s − 27-s − 2.23·28-s + ⋯ |
| L(s) = 1 | + 0.831·2-s − 0.577·3-s − 0.307·4-s + 1.27·5-s − 0.480·6-s + 1.37·7-s − 1.08·8-s + 0.333·9-s + 1.06·10-s + 0.421·11-s + 0.177·12-s + 0.564·13-s + 1.14·14-s − 0.736·15-s − 0.597·16-s − 0.0622·17-s + 0.277·18-s + 0.365·19-s − 0.392·20-s − 0.792·21-s + 0.351·22-s + 0.208·23-s + 0.628·24-s + 0.628·25-s + 0.470·26-s − 0.192·27-s − 0.422·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.925206347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925206347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 0.256T + 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 5.80T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 + 4.00T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 - 1.95T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 8.14T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.27T + 83T^{2} \) |
| 89 | \( 1 - 3.67T + 89T^{2} \) |
| 97 | \( 1 - 8.10T + 97T^{2} \) |
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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
−9.147124098818999654461171283719, −8.538420761950143894327120994946, −7.48150320924697286033868070522, −6.33853378226220839733544603190, −5.85539755293590235787723796420, −5.07803158257385112485349097593, −4.57975812217268335381752934937, −3.50924019228770886662654474058, −2.16744966255778756004471386124, −1.15457992768515304131718986247,
1.15457992768515304131718986247, 2.16744966255778756004471386124, 3.50924019228770886662654474058, 4.57975812217268335381752934937, 5.07803158257385112485349097593, 5.85539755293590235787723796420, 6.33853378226220839733544603190, 7.48150320924697286033868070522, 8.538420761950143894327120994946, 9.147124098818999654461171283719
