| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s + 12-s + 6·13-s + 14-s − 2·15-s + 16-s + 17-s − 18-s − 2·20-s − 21-s − 8·23-s − 24-s − 25-s − 6·26-s + 27-s − 28-s + 6·29-s + 2·30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.447·20-s − 0.218·21-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.365·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075858513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075858513\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.70794133258610, −12.21713771986754, −11.84357035005775, −11.57130724016927, −10.79645438138171, −10.37887647286531, −10.22822131607696, −9.435915432047634, −9.058129697283077, −8.513469303690342, −8.196784492167157, −7.872343195310616, −7.336799360140015, −6.728875360644758, −6.302983707934138, −5.849841444254471, −5.245188113070527, −4.241641906568288, −4.000697418406772, −3.631802005353173, −2.880209948354235, −2.463315036410687, −1.688210377917721, −1.013482199951452, −0.4972747967542780,
0.4972747967542780, 1.013482199951452, 1.688210377917721, 2.463315036410687, 2.880209948354235, 3.631802005353173, 4.000697418406772, 4.241641906568288, 5.245188113070527, 5.849841444254471, 6.302983707934138, 6.728875360644758, 7.336799360140015, 7.872343195310616, 8.196784492167157, 8.513469303690342, 9.058129697283077, 9.435915432047634, 10.22822131607696, 10.37887647286531, 10.79645438138171, 11.57130724016927, 11.84357035005775, 12.21713771986754, 12.70794133258610
