L(s) = 1 | − 0.971·2-s − 2.84·3-s − 1.05·4-s − 2.58·5-s + 2.76·6-s − 3.41·7-s + 2.96·8-s + 5.10·9-s + 2.50·10-s + 2.01·11-s + 3.00·12-s − 0.0969·13-s + 3.32·14-s + 7.35·15-s − 0.774·16-s − 1.09·17-s − 4.96·18-s + 6.25·19-s + 2.72·20-s + 9.72·21-s − 1.95·22-s − 2.34·23-s − 8.45·24-s + 1.66·25-s + 0.0942·26-s − 5.99·27-s + 3.60·28-s + ⋯ |
L(s) = 1 | − 0.687·2-s − 1.64·3-s − 0.527·4-s − 1.15·5-s + 1.12·6-s − 1.29·7-s + 1.04·8-s + 1.70·9-s + 0.793·10-s + 0.607·11-s + 0.867·12-s − 0.0269·13-s + 0.887·14-s + 1.89·15-s − 0.193·16-s − 0.265·17-s − 1.16·18-s + 1.43·19-s + 0.609·20-s + 2.12·21-s − 0.417·22-s − 0.488·23-s − 1.72·24-s + 0.333·25-s + 0.0184·26-s − 1.15·27-s + 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
good | 2 | \( 1 + 0.971T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 0.0969T + 13T^{2} \) |
| 17 | \( 1 + 1.09T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 8.08T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 + 8.98T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 0.780T + 61T^{2} \) |
| 67 | \( 1 - 8.13T + 67T^{2} \) |
| 71 | \( 1 - 7.51T + 71T^{2} \) |
| 73 | \( 1 + 3.98T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−9.838467286131611172409690784527, −8.865326146830561421845412595044, −7.82845618612143930496635360486, −6.98768356803456915537669675355, −6.30212791578243430006601847936, −5.21029463852273574259919118734, −4.33549263071831295202979472937, −3.45160803423485099653283559179, −0.973401083534424954556142204296, 0,
0.973401083534424954556142204296, 3.45160803423485099653283559179, 4.33549263071831295202979472937, 5.21029463852273574259919118734, 6.30212791578243430006601847936, 6.98768356803456915537669675355, 7.82845618612143930496635360486, 8.865326146830561421845412595044, 9.838467286131611172409690784527