| L(s) = 1 | − 2.63·2-s + 1.40·3-s + 4.91·4-s − 3.68·6-s − 1.05·7-s − 7.67·8-s − 1.03·9-s + 0.340·11-s + 6.88·12-s − 6.67·13-s + 2.77·14-s + 10.3·16-s − 1.88·17-s + 2.73·18-s − 0.838·19-s − 1.47·21-s − 0.894·22-s − 0.613·23-s − 10.7·24-s + 17.5·26-s − 5.65·27-s − 5.18·28-s + 1.63·29-s + 6.39·31-s − 11.8·32-s + 0.476·33-s + 4.95·34-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.808·3-s + 2.45·4-s − 1.50·6-s − 0.398·7-s − 2.71·8-s − 0.346·9-s + 0.102·11-s + 1.98·12-s − 1.85·13-s + 0.741·14-s + 2.58·16-s − 0.456·17-s + 0.643·18-s − 0.192·19-s − 0.322·21-s − 0.190·22-s − 0.127·23-s − 2.19·24-s + 3.44·26-s − 1.08·27-s − 0.979·28-s + 0.303·29-s + 1.14·31-s − 2.09·32-s + 0.0829·33-s + 0.849·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5500175337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5500175337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.340T + 11T^{2} \) |
| 13 | \( 1 + 6.67T + 13T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 + 0.838T + 19T^{2} \) |
| 23 | \( 1 + 0.613T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 - 6.39T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 0.850T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 + 0.258T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.063253553930195035812599198528, −7.75128756095604630227350923662, −6.93299439329798184197012865481, −6.43030341641530839546102564615, −5.42997596089502043221607443829, −4.30595725743878606976512225280, −2.99880014743023229525942823316, −2.59688857860786000873104698073, −1.83137781757118167791607198175, −0.46636321422910279743335589875,
0.46636321422910279743335589875, 1.83137781757118167791607198175, 2.59688857860786000873104698073, 2.99880014743023229525942823316, 4.30595725743878606976512225280, 5.42997596089502043221607443829, 6.43030341641530839546102564615, 6.93299439329798184197012865481, 7.75128756095604630227350923662, 8.063253553930195035812599198528
