L(s) = 1 | − 2.41·2-s + 3.82·4-s + 5-s + 1.82·7-s − 4.41·8-s − 2.41·10-s − 3.82·11-s − 13-s − 4.41·14-s + 2.99·16-s − 4.65·17-s − 1.17·19-s + 3.82·20-s + 9.24·22-s + 5.65·23-s + 25-s + 2.41·26-s + 6.99·28-s − 5.65·29-s + 6·31-s + 1.58·32-s + 11.2·34-s + 1.82·35-s + 6·37-s + 2.82·38-s − 4.41·40-s − 0.171·41-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.447·5-s + 0.691·7-s − 1.56·8-s − 0.763·10-s − 1.15·11-s − 0.277·13-s − 1.17·14-s + 0.749·16-s − 1.12·17-s − 0.268·19-s + 0.856·20-s + 1.97·22-s + 1.17·23-s + 0.200·25-s + 0.473·26-s + 1.32·28-s − 1.05·29-s + 1.07·31-s + 0.280·32-s + 1.92·34-s + 0.309·35-s + 0.986·37-s + 0.458·38-s − 0.697·40-s − 0.0267·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7496360562\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7496360562\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 + 9.82T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−9.147894349373452450586386159259, −8.681903930146918689937762810875, −7.82753511948930887335260310493, −7.29630200039880482413779825164, −6.41250025057189420212642793264, −5.39241930339422735660709895139, −4.44095270112258870271960048009, −2.67532707506727818441252005091, −2.07767885707239788340661619185, −0.75616197381196889051848635290,
0.75616197381196889051848635290, 2.07767885707239788340661619185, 2.67532707506727818441252005091, 4.44095270112258870271960048009, 5.39241930339422735660709895139, 6.41250025057189420212642793264, 7.29630200039880482413779825164, 7.82753511948930887335260310493, 8.681903930146918689937762810875, 9.147894349373452450586386159259