L(s) = 1 | − 2-s − 2.73·3-s + 4-s + 5-s + 2.73·6-s − 0.880·7-s − 8-s + 4.49·9-s − 10-s + 2.88·11-s − 2.73·12-s + 13-s + 0.880·14-s − 2.73·15-s + 16-s − 2.60·17-s − 4.49·18-s − 3.20·19-s + 20-s + 2.41·21-s − 2.88·22-s + 2.33·23-s + 2.73·24-s + 25-s − 26-s − 4.10·27-s − 0.880·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s − 0.332·7-s − 0.353·8-s + 1.49·9-s − 0.316·10-s + 0.870·11-s − 0.790·12-s + 0.277·13-s + 0.235·14-s − 0.707·15-s + 0.250·16-s − 0.631·17-s − 1.06·18-s − 0.736·19-s + 0.223·20-s + 0.525·21-s − 0.615·22-s + 0.487·23-s + 0.558·24-s + 0.200·25-s − 0.196·26-s − 0.789·27-s − 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.880T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 + 0.826T + 29T^{2} \) |
| 37 | \( 1 + 9.57T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 8.96T + 43T^{2} \) |
| 47 | \( 1 + 8.62T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.57T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 8.09T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 4.56T + 97T^{2} \) |
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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
−8.078738018472922343897470913750, −7.06637789757142950118120778450, −6.45874065139813188900840060005, −6.17343263612869962342177277705, −5.24974406640901775677550972814, −4.48111811744317232394597228678, −3.43619936265758172393224459994, −2.06677484420629230808926127120, −1.12380967815255804719658173558, 0,
1.12380967815255804719658173558, 2.06677484420629230808926127120, 3.43619936265758172393224459994, 4.48111811744317232394597228678, 5.24974406640901775677550972814, 6.17343263612869962342177277705, 6.45874065139813188900840060005, 7.06637789757142950118120778450, 8.078738018472922343897470913750