| L(s) = 1 | + 2.14·2-s + 2.50·3-s + 2.60·4-s − 0.390·5-s + 5.38·6-s − 3.27·7-s + 1.30·8-s + 3.29·9-s − 0.838·10-s + 11-s + 6.54·12-s − 1.62·13-s − 7.03·14-s − 0.979·15-s − 2.41·16-s − 3.12·17-s + 7.06·18-s − 1.25·19-s − 1.01·20-s − 8.22·21-s + 2.14·22-s − 4.86·23-s + 3.28·24-s − 4.84·25-s − 3.49·26-s + 0.728·27-s − 8.55·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.44·3-s + 1.30·4-s − 0.174·5-s + 2.19·6-s − 1.23·7-s + 0.462·8-s + 1.09·9-s − 0.265·10-s + 0.301·11-s + 1.88·12-s − 0.451·13-s − 1.88·14-s − 0.252·15-s − 0.602·16-s − 0.757·17-s + 1.66·18-s − 0.287·19-s − 0.227·20-s − 1.79·21-s + 0.457·22-s − 1.01·23-s + 0.669·24-s − 0.969·25-s − 0.684·26-s + 0.140·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 + 0.390T + 5T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 - 1.48T + 97T^{2} \) |
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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
−7.28536734001735145079344307677, −6.40484103856293619741423250522, −6.20177757339586868717777326734, −5.09993338006520751593758885111, −4.31200008211792789594564292739, −3.70611188975407482792849677543, −3.30020947463470961526338336504, −2.49435746814589570674801191286, −1.97101795680738641828636910648, 0,
1.97101795680738641828636910648, 2.49435746814589570674801191286, 3.30020947463470961526338336504, 3.70611188975407482792849677543, 4.31200008211792789594564292739, 5.09993338006520751593758885111, 6.20177757339586868717777326734, 6.40484103856293619741423250522, 7.28536734001735145079344307677
