L(s) = 1 | − 3-s − 7-s − 2·9-s + 5·11-s − 13-s − 3·17-s + 6·19-s + 21-s − 6·23-s + 5·27-s − 9·29-s − 5·33-s − 6·37-s + 39-s + 8·41-s + 6·43-s + 3·47-s + 49-s + 3·51-s + 12·53-s − 6·57-s − 8·59-s − 4·61-s + 2·63-s − 4·67-s + 6·69-s − 8·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s − 0.727·17-s + 1.37·19-s + 0.218·21-s − 1.25·23-s + 0.962·27-s − 1.67·29-s − 0.870·33-s − 0.986·37-s + 0.160·39-s + 1.24·41-s + 0.914·43-s + 0.437·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.794·57-s − 1.04·59-s − 0.512·61-s + 0.251·63-s − 0.488·67-s + 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.589904758444359152323012508760, −7.46356808305327002580621452051, −6.91794720272875152444408342798, −5.88365067292259041824349367828, −5.67666361705912888987123505751, −4.39488442217543021925706257311, −3.71321502126357141767462189683, −2.64244161461415544715262623018, −1.38820602899534117327773555596, 0,
1.38820602899534117327773555596, 2.64244161461415544715262623018, 3.71321502126357141767462189683, 4.39488442217543021925706257311, 5.67666361705912888987123505751, 5.88365067292259041824349367828, 6.91794720272875152444408342798, 7.46356808305327002580621452051, 8.589904758444359152323012508760