| L(s) = 1 | + 0.339·2-s − 3-s − 1.88·4-s − 5-s − 0.339·6-s − 0.660·7-s − 1.32·8-s + 9-s − 0.339·10-s − 0.679·11-s + 1.88·12-s − 0.224·14-s + 15-s + 3.32·16-s + 7.42·17-s + 0.339·18-s − 0.115·19-s + 1.88·20-s + 0.660·21-s − 0.231·22-s − 7.76·23-s + 1.32·24-s + 25-s − 27-s + 1.24·28-s + 5.54·29-s + 0.339·30-s + ⋯ |
| L(s) = 1 | + 0.240·2-s − 0.577·3-s − 0.942·4-s − 0.447·5-s − 0.138·6-s − 0.249·7-s − 0.466·8-s + 0.333·9-s − 0.107·10-s − 0.204·11-s + 0.544·12-s − 0.0599·14-s + 0.258·15-s + 0.830·16-s + 1.80·17-s + 0.0801·18-s − 0.0265·19-s + 0.421·20-s + 0.144·21-s − 0.0492·22-s − 1.61·23-s + 0.269·24-s + 0.200·25-s − 0.192·27-s + 0.235·28-s + 1.02·29-s + 0.0620·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.339T + 2T^{2} \) |
| 7 | \( 1 + 0.660T + 7T^{2} \) |
| 11 | \( 1 + 0.679T + 11T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 + 0.115T + 19T^{2} \) |
| 23 | \( 1 + 7.76T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 9.97T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 + 0.544T + 43T^{2} \) |
| 47 | \( 1 - 5.01T + 47T^{2} \) |
| 53 | \( 1 - 0.679T + 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 + 1.76T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 9.90T + 97T^{2} \) |
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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
−8.235517357906576023342250645686, −8.055579378049151206181560362436, −6.89710669684316323689596444264, −6.01594598271490642684742414682, −5.35301096090062668599669603065, −4.57799276765654370971995329772, −3.78115364673898168714355200209, −2.96503395626382682106119639552, −1.22814428007368003786265667405, 0,
1.22814428007368003786265667405, 2.96503395626382682106119639552, 3.78115364673898168714355200209, 4.57799276765654370971995329772, 5.35301096090062668599669603065, 6.01594598271490642684742414682, 6.89710669684316323689596444264, 8.055579378049151206181560362436, 8.235517357906576023342250645686
