| L(s) = 1 | − 1.14·3-s − 2.48·5-s + 0.657·7-s − 1.68·9-s − 2.68·11-s + 0.853·13-s + 2.85·15-s − 0.853·17-s − 6.02·19-s − 0.753·21-s + 5.14·23-s + 1.19·25-s + 5.37·27-s − 3.53·29-s − 31-s + 3.07·33-s − 1.63·35-s − 2·37-s − 0.978·39-s − 3.80·41-s + 2.12·43-s + 4.19·45-s + 11.0·47-s − 6.56·49-s + 0.978·51-s + 6.97·53-s + 6.68·55-s + ⋯ |
| L(s) = 1 | − 0.661·3-s − 1.11·5-s + 0.248·7-s − 0.561·9-s − 0.809·11-s + 0.236·13-s + 0.736·15-s − 0.207·17-s − 1.38·19-s − 0.164·21-s + 1.07·23-s + 0.239·25-s + 1.03·27-s − 0.657·29-s − 0.179·31-s + 0.535·33-s − 0.276·35-s − 0.328·37-s − 0.156·39-s − 0.593·41-s + 0.324·43-s + 0.625·45-s + 1.61·47-s − 0.938·49-s + 0.137·51-s + 0.958·53-s + 0.901·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6414733740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6414733740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 0.657T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 0.853T + 13T^{2} \) |
| 17 | \( 1 + 0.853T + 17T^{2} \) |
| 19 | \( 1 + 6.02T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.95T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−8.852383268616067289850304053191, −8.457798659940327938346083860357, −7.59040994414595899462248485005, −6.86998346178916997677234059911, −5.90936079365376446192475186975, −5.14218180231278819500906731271, −4.32824122391031321351311246287, −3.40111363734765533078391647525, −2.26891748417863144442329618994, −0.52784643186119827166045527878,
0.52784643186119827166045527878, 2.26891748417863144442329618994, 3.40111363734765533078391647525, 4.32824122391031321351311246287, 5.14218180231278819500906731271, 5.90936079365376446192475186975, 6.86998346178916997677234059911, 7.59040994414595899462248485005, 8.457798659940327938346083860357, 8.852383268616067289850304053191
