L(s) = 1 | + 1.41·3-s + 5-s − 7-s − 0.999·9-s + 11-s − 3.41·13-s + 1.41·15-s + 2.24·17-s − 5.65·19-s − 1.41·21-s + 2·23-s + 25-s − 5.65·27-s − 3.17·29-s + 4.58·31-s + 1.41·33-s − 35-s − 2.82·37-s − 4.82·39-s + 4.24·41-s + 2·43-s − 0.999·45-s + 8.24·47-s + 49-s + 3.17·51-s − 4.48·53-s + 55-s + ⋯ |
L(s) = 1 | + 0.816·3-s + 0.447·5-s − 0.377·7-s − 0.333·9-s + 0.301·11-s − 0.946·13-s + 0.365·15-s + 0.543·17-s − 1.29·19-s − 0.308·21-s + 0.417·23-s + 0.200·25-s − 1.08·27-s − 0.588·29-s + 0.823·31-s + 0.246·33-s − 0.169·35-s − 0.464·37-s − 0.773·39-s + 0.662·41-s + 0.304·43-s − 0.149·45-s + 1.20·47-s + 0.142·49-s + 0.444·51-s − 0.616·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.17T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−7.68688458720070061845835381524, −7.15358351546254101976085321244, −6.20664932831447427557819146355, −5.71873819111444741285944103849, −4.71453486660785567636391838404, −3.97639587985772572970906002800, −2.97704764759380362073760594718, −2.52275961003715365304911276551, −1.52916152982208687347970503711, 0,
1.52916152982208687347970503711, 2.52275961003715365304911276551, 2.97704764759380362073760594718, 3.97639587985772572970906002800, 4.71453486660785567636391838404, 5.71873819111444741285944103849, 6.20664932831447427557819146355, 7.15358351546254101976085321244, 7.68688458720070061845835381524