| L(s) = 1 | − 3.60·3-s − 2.21·5-s − 4.42·7-s − 14.0·9-s − 12.3·11-s + 48.6·13-s + 7.97·15-s − 101.·17-s − 59.2·19-s + 15.9·21-s − 18.0·23-s − 120.·25-s + 147.·27-s + 29·29-s − 20.8·31-s + 44.6·33-s + 9.77·35-s − 101.·37-s − 175.·39-s + 40.7·41-s + 152.·43-s + 30.9·45-s − 121.·47-s − 323.·49-s + 365.·51-s − 177.·53-s + 27.4·55-s + ⋯ |
| L(s) = 1 | − 0.693·3-s − 0.197·5-s − 0.238·7-s − 0.518·9-s − 0.339·11-s + 1.03·13-s + 0.137·15-s − 1.44·17-s − 0.715·19-s + 0.165·21-s − 0.163·23-s − 0.960·25-s + 1.05·27-s + 0.185·29-s − 0.120·31-s + 0.235·33-s + 0.0472·35-s − 0.449·37-s − 0.719·39-s + 0.155·41-s + 0.540·43-s + 0.102·45-s − 0.376·47-s − 0.942·49-s + 1.00·51-s − 0.460·53-s + 0.0671·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6247959940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6247959940\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 3.60T + 27T^{2} \) |
| 5 | \( 1 + 2.21T + 125T^{2} \) |
| 7 | \( 1 + 4.42T + 343T^{2} \) |
| 11 | \( 1 + 12.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.0T + 1.21e4T^{2} \) |
| 31 | \( 1 + 20.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 40.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 152.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 109.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 61.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 471.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 169.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−8.717241747905845761285561613447, −8.287680416233651716697518654001, −7.17154985178421822366248165565, −6.28811726016516622683607964524, −5.88540284967537646962097905030, −4.81208146739665162395787704916, −4.00845322439387898969074545875, −2.94119380656423376600886343846, −1.81073669264476657940451618591, −0.37007931889644157044188192798,
0.37007931889644157044188192798, 1.81073669264476657940451618591, 2.94119380656423376600886343846, 4.00845322439387898969074545875, 4.81208146739665162395787704916, 5.88540284967537646962097905030, 6.28811726016516622683607964524, 7.17154985178421822366248165565, 8.287680416233651716697518654001, 8.717241747905845761285561613447
