| L(s) = 1 | − 1.12·2-s − 0.732·4-s − 5-s + 0.606·7-s + 3.07·8-s + 1.12·10-s + 3.07·11-s − 0.682·14-s − 2.00·16-s − 1.95·17-s + 4.88·19-s + 0.732·20-s − 3.46·22-s − 1.86·23-s + 25-s − 0.443·28-s + 2.78·29-s − 9.15·31-s − 3.90·32-s + 2.19·34-s − 0.606·35-s − 3.74·37-s − 5.50·38-s − 3.07·40-s − 1.07·41-s − 5.01·43-s − 2.25·44-s + ⋯ |
| L(s) = 1 | − 0.796·2-s − 0.366·4-s − 0.447·5-s + 0.229·7-s + 1.08·8-s + 0.356·10-s + 0.927·11-s − 0.182·14-s − 0.500·16-s − 0.473·17-s + 1.12·19-s + 0.163·20-s − 0.738·22-s − 0.388·23-s + 0.200·25-s − 0.0838·28-s + 0.516·29-s − 1.64·31-s − 0.689·32-s + 0.376·34-s − 0.102·35-s − 0.616·37-s − 0.892·38-s − 0.486·40-s − 0.167·41-s − 0.764·43-s − 0.339·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 7 | \( 1 - 0.606T + 7T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 + 1.97T + 67T^{2} \) |
| 71 | \( 1 + 5.90T + 71T^{2} \) |
| 73 | \( 1 + 4.35T + 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 + 6.97T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.42674420832841549798692525871, −7.27848827519241524394411736625, −6.24273854337861497641052463142, −5.36607640279771164103047389373, −4.62702560185648459182675907361, −3.95610129861326889772558763160, −3.23369821692786009585882604563, −1.90191016461503590174130563520, −1.13030520102778794161854977420, 0,
1.13030520102778794161854977420, 1.90191016461503590174130563520, 3.23369821692786009585882604563, 3.95610129861326889772558763160, 4.62702560185648459182675907361, 5.36607640279771164103047389373, 6.24273854337861497641052463142, 7.27848827519241524394411736625, 7.42674420832841549798692525871
