| L(s) = 1 | + 2.55·2-s − 2.08·3-s + 4.54·4-s − 5.33·6-s − 0.752·7-s + 6.51·8-s + 1.34·9-s − 6.22·11-s − 9.47·12-s + 6.48·13-s − 1.92·14-s + 7.56·16-s + 3.43·18-s − 6.05·19-s + 1.56·21-s − 15.9·22-s + 2.69·23-s − 13.5·24-s + 16.5·26-s + 3.45·27-s − 3.42·28-s + 5.99·29-s − 3.55·31-s + 6.33·32-s + 12.9·33-s + 6.10·36-s + 0.0501·37-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.20·3-s + 2.27·4-s − 2.17·6-s − 0.284·7-s + 2.30·8-s + 0.448·9-s − 1.87·11-s − 2.73·12-s + 1.79·13-s − 0.514·14-s + 1.89·16-s + 0.810·18-s − 1.38·19-s + 0.342·21-s − 3.39·22-s + 0.562·23-s − 2.77·24-s + 3.25·26-s + 0.664·27-s − 0.646·28-s + 1.11·29-s − 0.639·31-s + 1.12·32-s + 2.25·33-s + 1.01·36-s + 0.00823·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 7 | \( 1 + 0.752T + 7T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 - 0.0501T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 - 0.958T + 83T^{2} \) |
| 89 | \( 1 + 0.208T + 89T^{2} \) |
| 97 | \( 1 - 5.39T + 97T^{2} \) |
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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
−7.10368788425630025642526386535, −6.32729550404608064274786200173, −6.13855988785705281350921063197, −5.36580151461406453369987123954, −4.88918738133108394089550996581, −4.19996546146044729682771546944, −3.25670057591474131543503189290, −2.66872484846216778596662548545, −1.50846859909289502463433276786, 0,
1.50846859909289502463433276786, 2.66872484846216778596662548545, 3.25670057591474131543503189290, 4.19996546146044729682771546944, 4.88918738133108394089550996581, 5.36580151461406453369987123954, 6.13855988785705281350921063197, 6.32729550404608064274786200173, 7.10368788425630025642526386535
