| L(s) = 1 | − 2-s − 2.24·3-s + 4-s − 5-s + 2.24·6-s − 2.96·7-s − 8-s + 2.03·9-s + 10-s + 3.78·11-s − 2.24·12-s + 1.59·13-s + 2.96·14-s + 2.24·15-s + 16-s − 1.08·17-s − 2.03·18-s − 1.19·19-s − 20-s + 6.65·21-s − 3.78·22-s − 9.20·23-s + 2.24·24-s + 25-s − 1.59·26-s + 2.16·27-s − 2.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 0.447·5-s + 0.916·6-s − 1.12·7-s − 0.353·8-s + 0.678·9-s + 0.316·10-s + 1.14·11-s − 0.647·12-s + 0.441·13-s + 0.793·14-s + 0.579·15-s + 0.250·16-s − 0.263·17-s − 0.479·18-s − 0.275·19-s − 0.223·20-s + 1.45·21-s − 0.806·22-s − 1.91·23-s + 0.458·24-s + 0.200·25-s − 0.312·26-s + 0.416·27-s − 0.560·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 + 1.19T + 19T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 0.377T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 + 6.13T + 71T^{2} \) |
| 73 | \( 1 + 0.754T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 - 5.67T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.165639531424325988034285547134, −7.12417354740629773622923277189, −6.52931502007893308951480643127, −6.14414418622588569895748267744, −5.35831402127468791397287574897, −4.11662359793209497165230728875, −3.60379587935780582933898709563, −2.25100378865759688174797999899, −0.943273183984849138103240408056, 0,
0.943273183984849138103240408056, 2.25100378865759688174797999899, 3.60379587935780582933898709563, 4.11662359793209497165230728875, 5.35831402127468791397287574897, 6.14414418622588569895748267744, 6.52931502007893308951480643127, 7.12417354740629773622923277189, 8.165639531424325988034285547134
