L(s) = 1 | − 10·2-s − 28·4-s + 1.17e3·7-s + 1.56e3·8-s + 2.65e3·11-s + 1.11e4·13-s − 1.17e4·14-s − 1.20e4·16-s + 3.10e4·17-s + 3.03e4·19-s − 2.65e4·22-s + 3.27e4·23-s − 1.11e5·26-s − 3.27e4·28-s − 1.63e5·29-s + 1.36e5·31-s − 7.95e4·32-s − 3.10e5·34-s − 1.66e4·37-s − 3.03e5·38-s + 4.83e5·41-s + 1.41e5·43-s − 7.42e4·44-s − 3.27e5·46-s − 1.03e5·47-s + 5.45e5·49-s − 3.13e5·52-s + ⋯ |
L(s) = 1 | − 0.883·2-s − 0.218·4-s + 1.28·7-s + 1.07·8-s + 0.600·11-s + 1.41·13-s − 1.13·14-s − 0.733·16-s + 1.53·17-s + 1.01·19-s − 0.530·22-s + 0.561·23-s − 1.24·26-s − 0.282·28-s − 1.24·29-s + 0.821·31-s − 0.428·32-s − 1.35·34-s − 0.0540·37-s − 0.896·38-s + 1.09·41-s + 0.270·43-s − 0.131·44-s − 0.496·46-s − 0.145·47-s + 0.662·49-s − 0.308·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.857442140\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.857442140\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 5 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1170 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2650 T + p^{7} T^{2} \) |
| 13 | \( 1 - 860 p T + p^{7} T^{2} \) |
| 17 | \( 1 - 31070 T + p^{7} T^{2} \) |
| 19 | \( 1 - 30316 T + p^{7} T^{2} \) |
| 23 | \( 1 - 32760 T + p^{7} T^{2} \) |
| 29 | \( 1 + 163150 T + p^{7} T^{2} \) |
| 31 | \( 1 - 136188 T + p^{7} T^{2} \) |
| 37 | \( 1 + 16640 T + p^{7} T^{2} \) |
| 41 | \( 1 - 483200 T + p^{7} T^{2} \) |
| 43 | \( 1 - 141080 T + p^{7} T^{2} \) |
| 47 | \( 1 + 103240 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1950130 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2643350 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2820922 T + p^{7} T^{2} \) |
| 67 | \( 1 - 506220 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2890900 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2877290 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5717556 T + p^{7} T^{2} \) |
| 83 | \( 1 - 3790380 T + p^{7} T^{2} \) |
| 89 | \( 1 + 10564500 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2158130 T + p^{7} T^{2} \) |
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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
−10.97861556797155465612588434716, −9.846458301241903245311351265600, −8.980507450008903136312069530719, −8.096725913359239005370087394501, −7.44388234568116609440877897938, −5.82732131739986591307675281551, −4.72732164558773949335512602696, −3.52678647634174601862493783525, −1.50827814221873029889901605308, −0.968149300687189402217265168089,
0.968149300687189402217265168089, 1.50827814221873029889901605308, 3.52678647634174601862493783525, 4.72732164558773949335512602696, 5.82732131739986591307675281551, 7.44388234568116609440877897938, 8.096725913359239005370087394501, 8.980507450008903136312069530719, 9.846458301241903245311351265600, 10.97861556797155465612588434716