L(s) = 1 | − 1.61·2-s + 2.61·3-s + 0.618·4-s − 0.618·5-s − 4.23·6-s + 2.23·8-s + 3.85·9-s + 1.00·10-s + 2.23·11-s + 1.61·12-s + 3·13-s − 1.61·15-s − 4.85·16-s + 7.47·17-s − 6.23·18-s − 1.85·19-s − 0.381·20-s − 3.61·22-s + 2.61·23-s + 5.85·24-s − 4.61·25-s − 4.85·26-s + 2.23·27-s + 1.14·29-s + 2.61·30-s + 7.38·31-s + 3.38·32-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 1.51·3-s + 0.309·4-s − 0.276·5-s − 1.72·6-s + 0.790·8-s + 1.28·9-s + 0.316·10-s + 0.674·11-s + 0.467·12-s + 0.832·13-s − 0.417·15-s − 1.21·16-s + 1.81·17-s − 1.46·18-s − 0.425·19-s − 0.0854·20-s − 0.771·22-s + 0.545·23-s + 1.19·24-s − 0.923·25-s − 0.951·26-s + 0.430·27-s + 0.212·29-s + 0.477·30-s + 1.32·31-s + 0.597·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.714841661\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714841661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 43 | \( 1 + 9.94T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 - 15T + 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 + 5.56T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.979173908219833097796157929492, −8.373717155885574710489964394010, −8.028635028771748010798222393763, −7.26113324420639451125554182076, −6.34037825940922110337747113072, −4.96986321904285540696941499465, −3.82899753865493452672851720748, −3.30275508726194121211852468520, −1.97495888608341075472217053728, −1.05055021946268001399832260043,
1.05055021946268001399832260043, 1.97495888608341075472217053728, 3.30275508726194121211852468520, 3.82899753865493452672851720748, 4.96986321904285540696941499465, 6.34037825940922110337747113072, 7.26113324420639451125554182076, 8.028635028771748010798222393763, 8.373717155885574710489964394010, 8.979173908219833097796157929492