L(s) = 1 | − 2.57·2-s − 2.39·3-s + 4.62·4-s + 1.19·5-s + 6.15·6-s − 1.20·7-s − 6.74·8-s + 2.71·9-s − 3.06·10-s + 3.73·11-s − 11.0·12-s − 1.79·13-s + 3.08·14-s − 2.84·15-s + 8.11·16-s − 2.65·17-s − 6.98·18-s − 3.32·19-s + 5.50·20-s + 2.87·21-s − 9.59·22-s + 0.361·23-s + 16.1·24-s − 3.58·25-s + 4.62·26-s + 0.681·27-s − 5.54·28-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 1.38·3-s + 2.31·4-s + 0.532·5-s + 2.51·6-s − 0.453·7-s − 2.38·8-s + 0.904·9-s − 0.968·10-s + 1.12·11-s − 3.18·12-s − 0.498·13-s + 0.825·14-s − 0.734·15-s + 2.02·16-s − 0.644·17-s − 1.64·18-s − 0.761·19-s + 1.23·20-s + 0.626·21-s − 2.04·22-s + 0.0754·23-s + 3.29·24-s − 0.716·25-s + 0.906·26-s + 0.131·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 2.39T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 2.65T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 0.361T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.96T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 8.62T + 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 + 0.965T + 97T^{2} \) |
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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
−9.642233698767781798069656852165, −9.017567809057348246605787294118, −8.018294529783912993948332528847, −6.88595705529730693894436719318, −6.44285889963405495918632039872, −5.77874021379479364667272829915, −4.34649356914268462077940282871, −2.51548194845765378513365395689, −1.27550483929408857074147591642, 0,
1.27550483929408857074147591642, 2.51548194845765378513365395689, 4.34649356914268462077940282871, 5.77874021379479364667272829915, 6.44285889963405495918632039872, 6.88595705529730693894436719318, 8.018294529783912993948332528847, 9.017567809057348246605787294118, 9.642233698767781798069656852165