L(s) = 1 | + 2-s − 0.865·3-s + 4-s − 3.71·5-s − 0.865·6-s + 7-s + 8-s − 2.25·9-s − 3.71·10-s − 5.77·11-s − 0.865·12-s + 14-s + 3.21·15-s + 16-s − 0.212·17-s − 2.25·18-s + 2.13·19-s − 3.71·20-s − 0.865·21-s − 5.77·22-s + 2.47·23-s − 0.865·24-s + 8.77·25-s + 4.54·27-s + 28-s − 0.0985·29-s + 3.21·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.499·3-s + 0.5·4-s − 1.65·5-s − 0.353·6-s + 0.377·7-s + 0.353·8-s − 0.750·9-s − 1.17·10-s − 1.74·11-s − 0.249·12-s + 0.267·14-s + 0.829·15-s + 0.250·16-s − 0.0514·17-s − 0.530·18-s + 0.490·19-s − 0.829·20-s − 0.188·21-s − 1.23·22-s + 0.516·23-s − 0.176·24-s + 1.75·25-s + 0.874·27-s + 0.188·28-s − 0.0183·29-s + 0.586·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174675191\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174675191\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.865T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 17 | \( 1 + 0.212T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 0.0985T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 7.87T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 0.231T + 59T^{2} \) |
| 61 | \( 1 - 8.03T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 + 5.60T + 73T^{2} \) |
| 79 | \( 1 + 9.19T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.569441617289212722970508216048, −8.092439742388492209097261175667, −7.47472140701596795106152291980, −6.68601715430114001302709169113, −5.51661836912373138137553436808, −5.06570129007217209135121402678, −4.25926513291906714647039091232, −3.26369686674349691019353642045, −2.55379909943842339131457831989, −0.61440644578513138209677841650,
0.61440644578513138209677841650, 2.55379909943842339131457831989, 3.26369686674349691019353642045, 4.25926513291906714647039091232, 5.06570129007217209135121402678, 5.51661836912373138137553436808, 6.68601715430114001302709169113, 7.47472140701596795106152291980, 8.092439742388492209097261175667, 8.569441617289212722970508216048