| L(s) = 1 | − 2-s − 0.644·3-s + 4-s + 3.51·5-s + 0.644·6-s − 8-s − 2.58·9-s − 3.51·10-s − 1.59·11-s − 0.644·12-s − 1.32·13-s − 2.26·15-s + 16-s + 5.92·17-s + 2.58·18-s + 0.345·19-s + 3.51·20-s + 1.59·22-s − 8.44·23-s + 0.644·24-s + 7.37·25-s + 1.32·26-s + 3.59·27-s − 7.80·29-s + 2.26·30-s + 4.13·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.371·3-s + 0.5·4-s + 1.57·5-s + 0.262·6-s − 0.353·8-s − 0.861·9-s − 1.11·10-s − 0.480·11-s − 0.185·12-s − 0.368·13-s − 0.584·15-s + 0.250·16-s + 1.43·17-s + 0.609·18-s + 0.0791·19-s + 0.786·20-s + 0.339·22-s − 1.76·23-s + 0.131·24-s + 1.47·25-s + 0.260·26-s + 0.692·27-s − 1.44·29-s + 0.413·30-s + 0.742·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.644T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 - 0.345T + 19T^{2} \) |
| 23 | \( 1 + 8.44T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 2.45T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−7.63527308637031206990193730871, −6.76124870385618409050389791235, −5.96732744601067436500508832810, −5.63288467275514089309014318679, −5.12795820602778020378780034683, −3.77323985366963731634354960240, −2.76150998075528219659959310593, −2.16097412528749528623328631692, −1.27529928174913941658220442149, 0,
1.27529928174913941658220442149, 2.16097412528749528623328631692, 2.76150998075528219659959310593, 3.77323985366963731634354960240, 5.12795820602778020378780034683, 5.63288467275514089309014318679, 5.96732744601067436500508832810, 6.76124870385618409050389791235, 7.63527308637031206990193730871
