| L(s) = 1 | + 2.05·2-s − 0.485·3-s + 2.23·4-s + 5-s − 0.998·6-s + 0.485·8-s − 2.76·9-s + 2.05·10-s + 0.984·11-s − 1.08·12-s + 1.77·13-s − 0.485·15-s − 3.47·16-s − 2.28·17-s − 5.68·18-s − 3.32·19-s + 2.23·20-s + 2.02·22-s + 5.37·23-s − 0.235·24-s + 25-s + 3.65·26-s + 2.79·27-s − 8.76·29-s − 0.998·30-s − 31-s − 8.11·32-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.280·3-s + 1.11·4-s + 0.447·5-s − 0.407·6-s + 0.171·8-s − 0.921·9-s + 0.650·10-s + 0.296·11-s − 0.313·12-s + 0.492·13-s − 0.125·15-s − 0.868·16-s − 0.553·17-s − 1.34·18-s − 0.761·19-s + 0.499·20-s + 0.432·22-s + 1.12·23-s − 0.0480·24-s + 0.200·25-s + 0.717·26-s + 0.538·27-s − 1.62·29-s − 0.182·30-s − 0.179·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 3 | \( 1 + 0.485T + 3T^{2} \) |
| 11 | \( 1 - 0.984T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 + 7.25T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 0.259T + 67T^{2} \) |
| 71 | \( 1 + 9.98T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 + 0.156T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.13041546141683121574375134217, −6.55671189943602768272957244854, −5.88561582798326589653251606378, −5.51956700978430845559094896240, −4.70117122875682924884358455382, −4.06755732352122462301595318955, −3.20828600166432085253983904061, −2.58350165263468604294363038514, −1.59025606191627306500837412462, 0,
1.59025606191627306500837412462, 2.58350165263468604294363038514, 3.20828600166432085253983904061, 4.06755732352122462301595318955, 4.70117122875682924884358455382, 5.51956700978430845559094896240, 5.88561582798326589653251606378, 6.55671189943602768272957244854, 7.13041546141683121574375134217
