| L(s) = 1 | + 0.718·2-s + 2.92·3-s − 1.48·4-s + 3.34·5-s + 2.09·6-s − 4.97·7-s − 2.50·8-s + 5.54·9-s + 2.40·10-s − 5.33·11-s − 4.33·12-s + 4.26·13-s − 3.57·14-s + 9.78·15-s + 1.17·16-s − 17-s + 3.98·18-s − 2.95·19-s − 4.96·20-s − 14.5·21-s − 3.83·22-s − 7.31·24-s + 6.19·25-s + 3.06·26-s + 7.44·27-s + 7.37·28-s − 1.24·29-s + ⋯ |
| L(s) = 1 | + 0.507·2-s + 1.68·3-s − 0.742·4-s + 1.49·5-s + 0.857·6-s − 1.87·7-s − 0.884·8-s + 1.84·9-s + 0.759·10-s − 1.60·11-s − 1.25·12-s + 1.18·13-s − 0.954·14-s + 2.52·15-s + 0.292·16-s − 0.242·17-s + 0.939·18-s − 0.678·19-s − 1.11·20-s − 3.17·21-s − 0.816·22-s − 1.49·24-s + 1.23·25-s + 0.600·26-s + 1.43·27-s + 1.39·28-s − 0.230·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8993 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.718T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 41 | \( 1 - 2.72T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 + 8.84T + 47T^{2} \) |
| 53 | \( 1 + 8.12T + 53T^{2} \) |
| 59 | \( 1 + 5.55T + 59T^{2} \) |
| 61 | \( 1 + 5.77T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−7.46453665518869388681560404088, −6.53256218058362919078557082969, −5.99709471225154779075810765986, −5.43798219096717492658577856874, −4.39138393662735061151455568121, −3.62476540609885027045957318788, −2.95746505133268890351725251857, −2.63980804951686037758861193665, −1.62042310630122792059073539407, 0,
1.62042310630122792059073539407, 2.63980804951686037758861193665, 2.95746505133268890351725251857, 3.62476540609885027045957318788, 4.39138393662735061151455568121, 5.43798219096717492658577856874, 5.99709471225154779075810765986, 6.53256218058362919078557082969, 7.46453665518869388681560404088
