L(s) = 1 | + 0.732·3-s + 5-s − 7-s − 2.46·9-s − 11-s + 5.46·13-s + 0.732·15-s − 3.46·17-s − 0.732·19-s − 0.732·21-s − 4.73·23-s + 25-s − 4·27-s − 1.26·29-s + 4.92·31-s − 0.732·33-s − 35-s + 6.73·37-s + 4·39-s − 1.26·41-s − 8.92·43-s − 2.46·45-s + 49-s − 2.53·51-s − 1.26·53-s − 55-s − 0.535·57-s + ⋯ |
L(s) = 1 | + 0.422·3-s + 0.447·5-s − 0.377·7-s − 0.821·9-s − 0.301·11-s + 1.51·13-s + 0.189·15-s − 0.840·17-s − 0.167·19-s − 0.159·21-s − 0.986·23-s + 0.200·25-s − 0.769·27-s − 0.235·29-s + 0.885·31-s − 0.127·33-s − 0.169·35-s + 1.10·37-s + 0.640·39-s − 0.198·41-s − 1.36·43-s − 0.367·45-s + 0.142·49-s − 0.355·51-s − 0.174·53-s − 0.134·55-s − 0.0709·57-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 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.732T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.965762484179657327686935884460, −6.88642557700726943376445874367, −6.10298691425295301548717371524, −5.88344457161043946417612412765, −4.77241007100744029014277526235, −3.92546367621878995045179968519, −3.13939929255758950953012992557, −2.41541798603661094798636525370, −1.44461918039345720744270432699, 0,
1.44461918039345720744270432699, 2.41541798603661094798636525370, 3.13939929255758950953012992557, 3.92546367621878995045179968519, 4.77241007100744029014277526235, 5.88344457161043946417612412765, 6.10298691425295301548717371524, 6.88642557700726943376445874367, 7.965762484179657327686935884460