| L(s) = 1 | − 2.40·2-s − 0.841·3-s + 3.79·4-s + 4.15·5-s + 2.02·6-s − 4.31·8-s − 2.29·9-s − 9.99·10-s + 0.388·11-s − 3.19·12-s − 6.22·13-s − 3.49·15-s + 2.80·16-s + 4.73·17-s + 5.51·18-s + 6.71·19-s + 15.7·20-s − 0.934·22-s + 5.02·23-s + 3.63·24-s + 12.2·25-s + 14.9·26-s + 4.45·27-s − 3.84·29-s + 8.41·30-s + 2.71·31-s + 1.88·32-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.485·3-s + 1.89·4-s + 1.85·5-s + 0.826·6-s − 1.52·8-s − 0.764·9-s − 3.16·10-s + 0.117·11-s − 0.921·12-s − 1.72·13-s − 0.902·15-s + 0.701·16-s + 1.14·17-s + 1.30·18-s + 1.53·19-s + 3.52·20-s − 0.199·22-s + 1.04·23-s + 0.741·24-s + 2.45·25-s + 2.93·26-s + 0.856·27-s − 0.714·29-s + 1.53·30-s + 0.488·31-s + 0.333·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030272259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030272259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 + 0.841T + 3T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 11 | \( 1 - 0.388T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 - 0.596T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 - 5.33T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 - 0.864T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−8.107224044309460089610253962430, −7.35165798139841354548425037394, −6.89606007424831578175461992275, −5.93429228751131656385380395085, −5.51621234181048921505552694473, −4.85916476442948567018173613102, −2.89562629099672537562590025072, −2.58418240969337232101332616446, −1.49693227252297244588412565494, −0.74698466383227781901838523522,
0.74698466383227781901838523522, 1.49693227252297244588412565494, 2.58418240969337232101332616446, 2.89562629099672537562590025072, 4.85916476442948567018173613102, 5.51621234181048921505552694473, 5.93429228751131656385380395085, 6.89606007424831578175461992275, 7.35165798139841354548425037394, 8.107224044309460089610253962430
