| L(s) = 1 | − 0.816·2-s − 1.53·3-s − 1.33·4-s + 1.25·6-s − 5.03·7-s + 2.72·8-s − 0.633·9-s − 3.03·11-s + 2.05·12-s + 4.57·13-s + 4.11·14-s + 0.443·16-s + 1.07·17-s + 0.517·18-s + 19-s + 7.74·21-s + 2.47·22-s − 4.11·23-s − 4.18·24-s − 3.73·26-s + 5.58·27-s + 6.71·28-s − 1.07·29-s + 5.58·31-s − 5.80·32-s + 4.66·33-s − 0.879·34-s + ⋯ |
| L(s) = 1 | − 0.577·2-s − 0.888·3-s − 0.666·4-s + 0.512·6-s − 1.90·7-s + 0.962·8-s − 0.211·9-s − 0.914·11-s + 0.592·12-s + 1.26·13-s + 1.09·14-s + 0.110·16-s + 0.261·17-s + 0.121·18-s + 0.229·19-s + 1.68·21-s + 0.528·22-s − 0.857·23-s − 0.854·24-s − 0.732·26-s + 1.07·27-s + 1.26·28-s − 0.199·29-s + 1.00·31-s − 1.02·32-s + 0.812·33-s − 0.150·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3567225561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3567225561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.816T + 2T^{2} \) |
| 3 | \( 1 + 1.53T + 3T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 0.0947T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
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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
−10.63206226457689822574229513159, −10.23071942198849525938986948920, −9.292832878707652797997066644687, −8.495178541244963706394724501283, −7.37293913524197533799493645969, −6.13384615958499342549676490383, −5.67595031944522013313221045089, −4.21642320315638735892081658755, −3.02795612468554015068081966573, −0.60852986743194229708444574820,
0.60852986743194229708444574820, 3.02795612468554015068081966573, 4.21642320315638735892081658755, 5.67595031944522013313221045089, 6.13384615958499342549676490383, 7.37293913524197533799493645969, 8.495178541244963706394724501283, 9.292832878707652797997066644687, 10.23071942198849525938986948920, 10.63206226457689822574229513159
