| L(s) = 1 | + 2-s + 3-s + 4-s − 0.942·5-s + 6-s − 0.429·7-s + 8-s + 9-s − 0.942·10-s − 6.18·11-s + 12-s + 5.59·13-s − 0.429·14-s − 0.942·15-s + 16-s + 1.51·17-s + 18-s + 19-s − 0.942·20-s − 0.429·21-s − 6.18·22-s − 5.32·23-s + 24-s − 4.11·25-s + 5.59·26-s + 27-s − 0.429·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.421·5-s + 0.408·6-s − 0.162·7-s + 0.353·8-s + 0.333·9-s − 0.298·10-s − 1.86·11-s + 0.288·12-s + 1.55·13-s − 0.114·14-s − 0.243·15-s + 0.250·16-s + 0.367·17-s + 0.235·18-s + 0.229·19-s − 0.210·20-s − 0.0936·21-s − 1.31·22-s − 1.10·23-s + 0.204·24-s − 0.822·25-s + 1.09·26-s + 0.192·27-s − 0.0811·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.533411589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.533411589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.942T + 5T^{2} \) |
| 7 | \( 1 + 0.429T + 7T^{2} \) |
| 11 | \( 1 + 6.18T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 59 | \( 1 + 0.958T + 59T^{2} \) |
| 61 | \( 1 - 0.890T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 - 6.24T + 97T^{2} \) |
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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
−7.912433593702105453843325595759, −7.70578923522266757283466117077, −6.38057341925686243061609500772, −6.06400487659073620418707985879, −5.08537293280321273824223632142, −4.36860103107213048258657861327, −3.60252939451816976445269219945, −2.90745098700591962253681433401, −2.19388855197284914324859507633, −0.861597328478426437777696960375,
0.861597328478426437777696960375, 2.19388855197284914324859507633, 2.90745098700591962253681433401, 3.60252939451816976445269219945, 4.36860103107213048258657861327, 5.08537293280321273824223632142, 6.06400487659073620418707985879, 6.38057341925686243061609500772, 7.70578923522266757283466117077, 7.912433593702105453843325595759
