L(s) = 1 | − 2·5-s − 2·7-s + 6·11-s + 13-s + 2·17-s − 6·19-s − 25-s − 6·29-s − 6·31-s + 4·35-s − 2·37-s + 10·41-s + 8·43-s + 6·47-s − 3·49-s + 6·53-s − 12·55-s + 6·59-s + 10·61-s − 2·65-s + 2·67-s − 14·71-s − 14·73-s − 12·77-s − 4·79-s − 6·83-s − 4·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.80·11-s + 0.277·13-s + 0.485·17-s − 1.37·19-s − 1/5·25-s − 1.11·29-s − 1.07·31-s + 0.676·35-s − 0.328·37-s + 1.56·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.61·55-s + 0.781·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 1.66·71-s − 1.63·73-s − 1.36·77-s − 0.450·79-s − 0.658·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.29386243308050133603790345980, −7.06268672217270991544014324871, −6.06087571590209650968038420818, −5.73528425947734469064705069073, −4.25072780514619507384193593587, −4.03103171659912262243396009243, −3.39330762915940877886976239656, −2.25317738391111529282424940634, −1.17086804304237406226337001447, 0,
1.17086804304237406226337001447, 2.25317738391111529282424940634, 3.39330762915940877886976239656, 4.03103171659912262243396009243, 4.25072780514619507384193593587, 5.73528425947734469064705069073, 6.06087571590209650968038420818, 7.06268672217270991544014324871, 7.29386243308050133603790345980