L(s) = 1 | + 2-s + 1.15·3-s + 4-s + 4.00·5-s + 1.15·6-s + 4.22·7-s + 8-s − 1.66·9-s + 4.00·10-s + 3.61·11-s + 1.15·12-s − 4.65·13-s + 4.22·14-s + 4.63·15-s + 16-s + 0.0314·17-s − 1.66·18-s + 6.36·19-s + 4.00·20-s + 4.89·21-s + 3.61·22-s − 23-s + 1.15·24-s + 11.0·25-s − 4.65·26-s − 5.39·27-s + 4.22·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.668·3-s + 0.5·4-s + 1.79·5-s + 0.472·6-s + 1.59·7-s + 0.353·8-s − 0.553·9-s + 1.26·10-s + 1.08·11-s + 0.334·12-s − 1.29·13-s + 1.12·14-s + 1.19·15-s + 0.250·16-s + 0.00762·17-s − 0.391·18-s + 1.46·19-s + 0.896·20-s + 1.06·21-s + 0.770·22-s − 0.208·23-s + 0.236·24-s + 2.21·25-s − 0.912·26-s − 1.03·27-s + 0.798·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.904310998\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.904310998\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 - 0.0314T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.86501890241179593513104392292, −7.47309368789882436055768282718, −6.52528095551141514746502857760, −5.66629440469035641232148110410, −5.28944208352333857697769743026, −4.64920412608864827965154787958, −3.55587493399491820994421767838, −2.65743579061023252320617846312, −1.91534584635523589406858972910, −1.45682747665174096412636564734,
1.45682747665174096412636564734, 1.91534584635523589406858972910, 2.65743579061023252320617846312, 3.55587493399491820994421767838, 4.64920412608864827965154787958, 5.28944208352333857697769743026, 5.66629440469035641232148110410, 6.52528095551141514746502857760, 7.47309368789882436055768282718, 7.86501890241179593513104392292