| L(s) = 1 | − 2-s + 3-s + 4-s + 1.09·5-s − 6-s + 1.89·7-s − 8-s + 9-s − 1.09·10-s − 11-s + 12-s − 2.23·13-s − 1.89·14-s + 1.09·15-s + 16-s − 0.681·17-s − 18-s − 1.28·19-s + 1.09·20-s + 1.89·21-s + 22-s − 5.26·23-s − 24-s − 3.79·25-s + 2.23·26-s + 27-s + 1.89·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.490·5-s − 0.408·6-s + 0.715·7-s − 0.353·8-s + 0.333·9-s − 0.347·10-s − 0.301·11-s + 0.288·12-s − 0.619·13-s − 0.505·14-s + 0.283·15-s + 0.250·16-s − 0.165·17-s − 0.235·18-s − 0.294·19-s + 0.245·20-s + 0.413·21-s + 0.213·22-s − 1.09·23-s − 0.204·24-s − 0.759·25-s + 0.437·26-s + 0.192·27-s + 0.357·28-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 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 0.681T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 5.26T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 7.33T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 - 7.93T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 67 | \( 1 - 0.379T + 67T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 + 0.375T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.48T + 97T^{2} \) |
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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
−8.085198576165811513639488324046, −7.55549910207353138105446754486, −6.86500839903310081789038522285, −5.87198498237554028798472281164, −5.18929769235158963765205424765, −4.18825423089404932833624306828, −3.23746890373320476897455512096, −2.06348580908651858049709033091, −1.75947534261781835650324793464, 0,
1.75947534261781835650324793464, 2.06348580908651858049709033091, 3.23746890373320476897455512096, 4.18825423089404932833624306828, 5.18929769235158963765205424765, 5.87198498237554028798472281164, 6.86500839903310081789038522285, 7.55549910207353138105446754486, 8.085198576165811513639488324046
