| L(s) = 1 | + 2.84·3-s + 0.145·7-s + 5.10·9-s + 5.71·11-s + 5.24·13-s + 7.15·17-s − 19-s + 0.412·21-s + 0.622·23-s + 6.00·27-s − 5.46·29-s + 3.77·31-s + 16.2·33-s − 5.03·37-s + 14.9·39-s + 5.77·41-s + 3.32·43-s − 5.85·47-s − 6.97·49-s + 20.3·51-s − 6.97·53-s − 2.84·57-s − 9.09·59-s − 8.16·61-s + 0.740·63-s − 13.6·67-s + 1.77·69-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 0.0548·7-s + 1.70·9-s + 1.72·11-s + 1.45·13-s + 1.73·17-s − 0.229·19-s + 0.0901·21-s + 0.129·23-s + 1.15·27-s − 1.01·29-s + 0.678·31-s + 2.83·33-s − 0.827·37-s + 2.39·39-s + 0.901·41-s + 0.506·43-s − 0.854·47-s − 0.996·49-s + 2.85·51-s − 0.957·53-s − 0.377·57-s − 1.18·59-s − 1.04·61-s + 0.0933·63-s − 1.66·67-s + 0.213·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.205621369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.205621369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.84T + 3T^{2} \) |
| 7 | \( 1 - 0.145T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 7.15T + 17T^{2} \) |
| 23 | \( 1 - 0.622T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 + 5.03T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 + 8.16T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 - 4.98T + 97T^{2} \) |
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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
−7.84162142976651447957329705339, −7.54866668294768551656450972634, −6.43617692677609656253707425337, −6.07542822985148024163903986887, −4.85786834259819778390033923176, −3.85751300902908814077678678635, −3.59552663912504136115310666561, −2.90996476896736343228278892237, −1.62857762898533227919339308452, −1.27802663322557661767391640523,
1.27802663322557661767391640523, 1.62857762898533227919339308452, 2.90996476896736343228278892237, 3.59552663912504136115310666561, 3.85751300902908814077678678635, 4.85786834259819778390033923176, 6.07542822985148024163903986887, 6.43617692677609656253707425337, 7.54866668294768551656450972634, 7.84162142976651447957329705339
