| L(s) = 1 | + 1.72·2-s − 2.96·3-s + 0.982·4-s + 1.08·5-s − 5.11·6-s − 1.51·7-s − 1.75·8-s + 5.77·9-s + 1.87·10-s − 5.06·11-s − 2.90·12-s − 2.50·13-s − 2.61·14-s − 3.22·15-s − 4.99·16-s − 1.60·17-s + 9.96·18-s + 0.616·19-s + 1.06·20-s + 4.48·21-s − 8.74·22-s − 0.286·23-s + 5.20·24-s − 3.81·25-s − 4.31·26-s − 8.20·27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 1.70·3-s + 0.491·4-s + 0.486·5-s − 2.08·6-s − 0.572·7-s − 0.621·8-s + 1.92·9-s + 0.594·10-s − 1.52·11-s − 0.839·12-s − 0.693·13-s − 0.698·14-s − 0.832·15-s − 1.24·16-s − 0.389·17-s + 2.34·18-s + 0.141·19-s + 0.239·20-s + 0.978·21-s − 1.86·22-s − 0.0598·23-s + 1.06·24-s − 0.763·25-s − 0.846·26-s − 1.57·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 401 | \( 1 + T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 - 0.616T + 19T^{2} \) |
| 23 | \( 1 + 0.286T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.65T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 - 0.372T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 4.14T + 97T^{2} \) |
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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
−11.01574953680648948587424943815, −10.24150631684685706160451950323, −9.346073490248882113654580912980, −7.55186304350611020327377865412, −6.48889185824238450965397081093, −5.68758413180960601444393801990, −5.19299082737600597275969364068, −4.20794305240342618901209651915, −2.58492994415211244658888385157, 0,
2.58492994415211244658888385157, 4.20794305240342618901209651915, 5.19299082737600597275969364068, 5.68758413180960601444393801990, 6.48889185824238450965397081093, 7.55186304350611020327377865412, 9.346073490248882113654580912980, 10.24150631684685706160451950323, 11.01574953680648948587424943815
