| L(s) = 1 | + 2.53·2-s + 4.41·4-s + 0.879·5-s + 6.10·8-s + 2.22·10-s + 3.87·11-s + 5.45·13-s + 6.63·16-s − 1.65·17-s − 2.41·19-s + 3.87·20-s + 9.82·22-s + 3.16·23-s − 4.22·25-s + 13.8·26-s − 6.04·29-s + 4.55·31-s + 4.59·32-s − 4.18·34-s − 4.55·37-s − 6.10·38-s + 5.36·40-s + 1.18·41-s + 0.184·43-s + 17.1·44-s + 8.00·46-s + 1.02·47-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s + 0.393·5-s + 2.15·8-s + 0.704·10-s + 1.16·11-s + 1.51·13-s + 1.65·16-s − 0.400·17-s − 0.553·19-s + 0.867·20-s + 2.09·22-s + 0.659·23-s − 0.845·25-s + 2.70·26-s − 1.12·29-s + 0.817·31-s + 0.812·32-s − 0.717·34-s − 0.748·37-s − 0.990·38-s + 0.849·40-s + 0.185·41-s + 0.0281·43-s + 2.58·44-s + 1.18·46-s + 0.149·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.223200111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.223200111\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 0.184T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 - 2.59T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 - 0.218T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.455391385490451886960149275162, −7.30394943554587703443441544490, −6.62139691538858663366397536556, −6.07424482951053050998266199482, −5.55812349492244493377850539477, −4.53173475166533445711632021729, −3.91322442388454155614814699402, −3.33510686307247030728745468715, −2.20909402578530652458598324714, −1.37441126382271694925582723902,
1.37441126382271694925582723902, 2.20909402578530652458598324714, 3.33510686307247030728745468715, 3.91322442388454155614814699402, 4.53173475166533445711632021729, 5.55812349492244493377850539477, 6.07424482951053050998266199482, 6.62139691538858663366397536556, 7.30394943554587703443441544490, 8.455391385490451886960149275162
