| L(s) = 1 | + 2.97·3-s − 1.84·5-s − 3.45·7-s + 5.87·9-s + 5.07·11-s − 1.93·13-s − 5.49·15-s − 17-s + 3.92·19-s − 10.2·21-s − 5.46·23-s − 1.59·25-s + 8.55·27-s − 0.561·29-s + 6.88·31-s + 15.1·33-s + 6.38·35-s + 8.43·37-s − 5.77·39-s − 3.95·41-s + 5.48·43-s − 10.8·45-s + 6.66·47-s + 4.94·49-s − 2.97·51-s − 12.2·53-s − 9.37·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s − 0.825·5-s − 1.30·7-s + 1.95·9-s + 1.53·11-s − 0.537·13-s − 1.41·15-s − 0.242·17-s + 0.900·19-s − 2.24·21-s − 1.13·23-s − 0.318·25-s + 1.64·27-s − 0.104·29-s + 1.23·31-s + 2.63·33-s + 1.07·35-s + 1.38·37-s − 0.924·39-s − 0.617·41-s + 0.836·43-s − 1.61·45-s + 0.971·47-s + 0.706·49-s − 0.417·51-s − 1.67·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083824474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083824474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 - 5.48T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.29T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 - 6.65T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.904178384108622972527291278329, −7.30871141841442884990664247097, −6.63621045584082292651604342419, −6.03244318943379102994335468873, −4.61385072579034812884580070071, −3.92419631260102163789732822835, −3.55546278938797978367048641641, −2.83929979223988446015231139610, −2.02941273031045654105943893989, −0.793978187189209969968584025645,
0.793978187189209969968584025645, 2.02941273031045654105943893989, 2.83929979223988446015231139610, 3.55546278938797978367048641641, 3.92419631260102163789732822835, 4.61385072579034812884580070071, 6.03244318943379102994335468873, 6.63621045584082292651604342419, 7.30871141841442884990664247097, 7.904178384108622972527291278329
