L(s) = 1 | − 6.87·2-s + 5.75·3-s + 15.2·4-s − 39.5·6-s + 159.·7-s + 115.·8-s − 209.·9-s − 66.6·11-s + 87.7·12-s − 3.15·13-s − 1.09e3·14-s − 1.27e3·16-s + 1.88e3·17-s + 1.44e3·18-s + 1.81e3·19-s + 920.·21-s + 458.·22-s + 2.03e3·23-s + 662.·24-s + 21.6·26-s − 2.60e3·27-s + 2.44e3·28-s − 5.05e3·29-s − 469.·31-s + 5.11e3·32-s − 383.·33-s − 1.29e4·34-s + ⋯ |
L(s) = 1 | − 1.21·2-s + 0.369·3-s + 0.476·4-s − 0.448·6-s + 1.23·7-s + 0.635·8-s − 0.863·9-s − 0.166·11-s + 0.176·12-s − 0.00516·13-s − 1.49·14-s − 1.24·16-s + 1.58·17-s + 1.04·18-s + 1.15·19-s + 0.455·21-s + 0.201·22-s + 0.802·23-s + 0.234·24-s + 0.00628·26-s − 0.687·27-s + 0.588·28-s − 1.11·29-s − 0.0876·31-s + 0.882·32-s − 0.0613·33-s − 1.92·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.578026987\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578026987\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
good | 2 | \( 1 + 6.87T + 32T^{2} \) |
| 3 | \( 1 - 5.75T + 243T^{2} \) |
| 7 | \( 1 - 159.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 66.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 3.15T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.88e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 469.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.18e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 491.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.18e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.29e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.60e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.82e4T + 8.58e9T^{2} \) |
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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
−9.092815827727739599994166205770, −8.306446241657040054825745327865, −7.79844596656850709069079965478, −7.19948741218049886625758768807, −5.62162292415828923722043155205, −5.04755138781048143210386755001, −3.73360337977605637871403077176, −2.57944792069428019412413649042, −1.47005034758242845904699169901, −0.70108425245302977320723813301,
0.70108425245302977320723813301, 1.47005034758242845904699169901, 2.57944792069428019412413649042, 3.73360337977605637871403077176, 5.04755138781048143210386755001, 5.62162292415828923722043155205, 7.19948741218049886625758768807, 7.79844596656850709069079965478, 8.306446241657040054825745327865, 9.092815827727739599994166205770