| L(s) = 1 | + 0.656·2-s − 0.204·3-s − 1.56·4-s + 1.35·5-s − 0.134·6-s − 2.34·8-s − 2.95·9-s + 0.892·10-s + 1.90·11-s + 0.321·12-s − 0.278·15-s + 1.60·16-s + 3.56·17-s − 1.94·18-s − 0.985·19-s − 2.13·20-s + 1.25·22-s + 1.69·23-s + 0.479·24-s − 3.15·25-s + 1.21·27-s + 6.54·29-s − 0.182·30-s − 7.69·31-s + 5.73·32-s − 0.390·33-s + 2.34·34-s + ⋯ |
| L(s) = 1 | + 0.463·2-s − 0.118·3-s − 0.784·4-s + 0.608·5-s − 0.0548·6-s − 0.828·8-s − 0.986·9-s + 0.282·10-s + 0.574·11-s + 0.0927·12-s − 0.0718·15-s + 0.400·16-s + 0.864·17-s − 0.457·18-s − 0.226·19-s − 0.477·20-s + 0.266·22-s + 0.353·23-s + 0.0978·24-s − 0.630·25-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s − 1.38·31-s + 1.01·32-s − 0.0679·33-s + 0.401·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 3 | \( 1 + 0.204T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 1.90T + 11T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 0.985T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 2.02T + 37T^{2} \) |
| 41 | \( 1 + 9.88T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 7.76T + 47T^{2} \) |
| 53 | \( 1 - 0.354T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 2.03T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.47471045594710939824721421469, −6.40692748606181296530646447718, −6.08317807152053092865562182046, −5.19065346780208609813190192519, −4.94660266588747611222586140547, −3.74938636099151667818091477826, −3.33537730490894149019626877718, −2.33577796029993186202790757595, −1.21399960076514160510021024523, 0,
1.21399960076514160510021024523, 2.33577796029993186202790757595, 3.33537730490894149019626877718, 3.74938636099151667818091477826, 4.94660266588747611222586140547, 5.19065346780208609813190192519, 6.08317807152053092865562182046, 6.40692748606181296530646447718, 7.47471045594710939824721421469
