| L(s) = 1 | + 2-s + 0.242·3-s + 4-s − 4.38·5-s + 0.242·6-s + 3.95·7-s + 8-s − 2.94·9-s − 4.38·10-s − 1.54·11-s + 0.242·12-s + 6.50·13-s + 3.95·14-s − 1.06·15-s + 16-s − 7.22·17-s − 2.94·18-s − 0.462·19-s − 4.38·20-s + 0.957·21-s − 1.54·22-s + 4.63·23-s + 0.242·24-s + 14.2·25-s + 6.50·26-s − 1.43·27-s + 3.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.139·3-s + 0.5·4-s − 1.96·5-s + 0.0988·6-s + 1.49·7-s + 0.353·8-s − 0.980·9-s − 1.38·10-s − 0.467·11-s + 0.0698·12-s + 1.80·13-s + 1.05·14-s − 0.274·15-s + 0.250·16-s − 1.75·17-s − 0.693·18-s − 0.106·19-s − 0.980·20-s + 0.209·21-s − 0.330·22-s + 0.965·23-s + 0.0494·24-s + 2.84·25-s + 1.27·26-s − 0.276·27-s + 0.747·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346727264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346727264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 0.242T + 3T^{2} \) |
| 5 | \( 1 + 4.38T + 5T^{2} \) |
| 7 | \( 1 - 3.95T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 6.50T + 13T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 19 | \( 1 + 0.462T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 - 0.697T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 - 3.45T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 + 3.54T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 + 9.32T + 89T^{2} \) |
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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
−8.119442241605268504989670246028, −7.40925856667953620591628521900, −6.79254697308282465524430347301, −5.77172493822571413368124517242, −5.01795215581182076712883620205, −4.33396399838095346884516949452, −3.81146704311472082469198765373, −3.04929472202640713847848373261, −2.00995479860966159575866405553, −0.71502615272735875146302645148,
0.71502615272735875146302645148, 2.00995479860966159575866405553, 3.04929472202640713847848373261, 3.81146704311472082469198765373, 4.33396399838095346884516949452, 5.01795215581182076712883620205, 5.77172493822571413368124517242, 6.79254697308282465524430347301, 7.40925856667953620591628521900, 8.119442241605268504989670246028
