| L(s) = 1 | − 0.554·2-s − 1.69·4-s − 1.37·5-s + 2.04·8-s + 0.765·10-s − 1.80·11-s − 4.20·13-s + 2.24·16-s + 4.47·17-s + 5.58·19-s + 2.33·20-s + 22-s + 0.0489·23-s − 3.09·25-s + 2.33·26-s + 3.15·29-s − 6.34·31-s − 5.34·32-s − 2.48·34-s + 3.18·37-s − 3.09·38-s − 2.82·40-s + 9.44·41-s + 2.45·43-s + 3.04·44-s − 0.0271·46-s + 9.10·47-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.846·4-s − 0.616·5-s + 0.724·8-s + 0.242·10-s − 0.543·11-s − 1.16·13-s + 0.561·16-s + 1.08·17-s + 1.28·19-s + 0.521·20-s + 0.213·22-s + 0.0101·23-s − 0.619·25-s + 0.457·26-s + 0.586·29-s − 1.14·31-s − 0.944·32-s − 0.426·34-s + 0.523·37-s − 0.502·38-s − 0.446·40-s + 1.47·41-s + 0.374·43-s + 0.459·44-s − 0.00400·46-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 - 0.0489T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 9.44T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + 0.954T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 + 0.273T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−8.154580202718377932645442935704, −7.63337373382167649493884995133, −7.32056490585352154054758511060, −5.83728028952741149212925639131, −5.21439930237808570205290243394, −4.44537255823646152222658773937, −3.60213056835530708345226261343, −2.65312714815011980343580339543, −1.16579496479026867121916089849, 0,
1.16579496479026867121916089849, 2.65312714815011980343580339543, 3.60213056835530708345226261343, 4.44537255823646152222658773937, 5.21439930237808570205290243394, 5.83728028952741149212925639131, 7.32056490585352154054758511060, 7.63337373382167649493884995133, 8.154580202718377932645442935704
