| L(s) = 1 | + 0.152·2-s − 1.31·3-s − 1.97·4-s + 5-s − 0.201·6-s − 0.959·7-s − 0.607·8-s − 1.26·9-s + 0.152·10-s + 11-s + 2.60·12-s − 2.82·13-s − 0.146·14-s − 1.31·15-s + 3.86·16-s + 4.22·17-s − 0.192·18-s − 2.07·19-s − 1.97·20-s + 1.26·21-s + 0.152·22-s − 3.32·23-s + 0.801·24-s + 25-s − 0.431·26-s + 5.61·27-s + 1.89·28-s + ⋯ |
| L(s) = 1 | + 0.108·2-s − 0.761·3-s − 0.988·4-s + 0.447·5-s − 0.0822·6-s − 0.362·7-s − 0.214·8-s − 0.420·9-s + 0.0483·10-s + 0.301·11-s + 0.752·12-s − 0.782·13-s − 0.0392·14-s − 0.340·15-s + 0.965·16-s + 1.02·17-s − 0.0454·18-s − 0.477·19-s − 0.441·20-s + 0.276·21-s + 0.0325·22-s − 0.692·23-s + 0.163·24-s + 0.200·25-s − 0.0845·26-s + 1.08·27-s + 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.152T + 2T^{2} \) |
| 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 + 0.959T + 7T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 6.69T + 53T^{2} \) |
| 59 | \( 1 - 9.63T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 1.25T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 5.78T + 83T^{2} \) |
| 89 | \( 1 - 3.42T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.273210145586807667745184582246, −7.28470324728122803264945965263, −6.30592258514175978326542641592, −5.86844657258231628199068457152, −5.07274294236732381111114947561, −4.51273587556191256183517008846, −3.47495967614372685283585939294, −2.58806069576320546776230128425, −1.10697777732000093400468939855, 0,
1.10697777732000093400468939855, 2.58806069576320546776230128425, 3.47495967614372685283585939294, 4.51273587556191256183517008846, 5.07274294236732381111114947561, 5.86844657258231628199068457152, 6.30592258514175978326542641592, 7.28470324728122803264945965263, 8.273210145586807667745184582246
