| L(s) = 1 | + 8.30·3-s + 29.0·7-s + 41.9·9-s + 33.7·11-s − 26.6·13-s + 74.5·17-s − 30.7·19-s + 241.·21-s + 177.·23-s + 124.·27-s + 57.3·29-s − 120.·31-s + 280.·33-s − 36.0·37-s − 221.·39-s + 54.4·41-s − 106.·43-s − 365.·47-s + 502.·49-s + 618.·51-s − 330.·53-s − 255.·57-s + 808.·59-s − 196.·61-s + 1.21e3·63-s + 274.·67-s + 1.47e3·69-s + ⋯ |
| L(s) = 1 | + 1.59·3-s + 1.57·7-s + 1.55·9-s + 0.925·11-s − 0.569·13-s + 1.06·17-s − 0.370·19-s + 2.50·21-s + 1.61·23-s + 0.884·27-s + 0.367·29-s − 0.700·31-s + 1.47·33-s − 0.160·37-s − 0.910·39-s + 0.207·41-s − 0.375·43-s − 1.13·47-s + 1.46·49-s + 1.69·51-s − 0.856·53-s − 0.592·57-s + 1.78·59-s − 0.412·61-s + 2.43·63-s + 0.500·67-s + 2.57·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.913323176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.913323176\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8.30T + 27T^{2} \) |
| 7 | \( 1 - 29.0T + 343T^{2} \) |
| 11 | \( 1 - 33.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 36.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 330.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 808.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 196.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 396.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 906.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 758.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 450.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 442.T + 9.12e5T^{2} \) |
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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
−8.694337294789381820782171160019, −8.137578861894785617966404927447, −7.48793512358672807150468766673, −6.78457481915737673228882246181, −5.36122626828445195325421147553, −4.61796124156765510364816681443, −3.73291435089094391405796394137, −2.87207855150519170081253569512, −1.86732096385355787064856056915, −1.17121737157443091098625306971,
1.17121737157443091098625306971, 1.86732096385355787064856056915, 2.87207855150519170081253569512, 3.73291435089094391405796394137, 4.61796124156765510364816681443, 5.36122626828445195325421147553, 6.78457481915737673228882246181, 7.48793512358672807150468766673, 8.137578861894785617966404927447, 8.694337294789381820782171160019
