L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 5.12·11-s + 12-s − 4·13-s + 14-s + 15-s + 16-s − 5.12·17-s − 18-s − 7.12·19-s + 20-s − 21-s − 5.12·22-s − 23-s − 24-s + 25-s + 4·26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.54·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.24·17-s − 0.235·18-s − 1.63·19-s + 0.223·20-s − 0.218·21-s − 1.09·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 + 6.87T + 97T^{2} \) |
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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
−8.182090315320464076958402657316, −7.06766625787066346828418025336, −6.61243017065147579625877436935, −6.15946769021603859048300893287, −4.76136648545871577061136626996, −4.18004987357755178407447061194, −3.08210481563735027122818368950, −2.24869472478024356431556343791, −1.51818504675068255596149514749, 0,
1.51818504675068255596149514749, 2.24869472478024356431556343791, 3.08210481563735027122818368950, 4.18004987357755178407447061194, 4.76136648545871577061136626996, 6.15946769021603859048300893287, 6.61243017065147579625877436935, 7.06766625787066346828418025336, 8.182090315320464076958402657316