| L(s) = 1 | − 2-s + 3-s + 4-s + 2.75·5-s − 6-s − 2.84·7-s − 8-s + 9-s − 2.75·10-s − 11-s + 12-s − 2.89·13-s + 2.84·14-s + 2.75·15-s + 16-s − 6.87·17-s − 18-s + 2.24·19-s + 2.75·20-s − 2.84·21-s + 22-s + 4.43·23-s − 24-s + 2.60·25-s + 2.89·26-s + 27-s − 2.84·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.23·5-s − 0.408·6-s − 1.07·7-s − 0.353·8-s + 0.333·9-s − 0.872·10-s − 0.301·11-s + 0.288·12-s − 0.802·13-s + 0.761·14-s + 0.712·15-s + 0.250·16-s − 1.66·17-s − 0.235·18-s + 0.516·19-s + 0.616·20-s − 0.621·21-s + 0.213·22-s + 0.923·23-s − 0.204·24-s + 0.521·25-s + 0.567·26-s + 0.192·27-s − 0.538·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 - 0.372T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 0.528T + 47T^{2} \) |
| 53 | \( 1 + 0.0665T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 + 0.778T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−8.196539545624058164149717838669, −7.26841741136997397523255130377, −6.77449618078237808675413596051, −6.03881077407887732384754948558, −5.24906402405690229715913122544, −4.18499852236459997149908716474, −2.93063895946999097653303213421, −2.50629131370288968119350801176, −1.54950363783201498041551492177, 0,
1.54950363783201498041551492177, 2.50629131370288968119350801176, 2.93063895946999097653303213421, 4.18499852236459997149908716474, 5.24906402405690229715913122544, 6.03881077407887732384754948558, 6.77449618078237808675413596051, 7.26841741136997397523255130377, 8.196539545624058164149717838669
