| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 4·11-s + 12-s − 14-s − 15-s − 16-s + 17-s − 18-s − 6·19-s − 20-s − 21-s − 4·22-s + 6·23-s − 3·24-s + 25-s − 27-s − 28-s + 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s + 1.25·23-s − 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s + 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.627865097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.627865097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.71291510720214, −12.20906292779591, −11.57349686313579, −11.32258170939039, −10.73755247843990, −10.39499907358583, −9.833346852000879, −9.570453885520236, −8.937218337141999, −8.515908228724402, −8.404171434744072, −7.427268841528141, −7.207976662069457, −6.597320693498391, −6.153638502394037, −5.627140408648917, −5.026955514240164, −4.623349696971307, −4.051543498074248, −3.773918398357193, −2.756099636815772, −2.207640405582058, −1.407557939651854, −1.106896009274790, −0.4687641132330538,
0.4687641132330538, 1.106896009274790, 1.407557939651854, 2.207640405582058, 2.756099636815772, 3.773918398357193, 4.051543498074248, 4.623349696971307, 5.026955514240164, 5.627140408648917, 6.153638502394037, 6.597320693498391, 7.207976662069457, 7.427268841528141, 8.404171434744072, 8.515908228724402, 8.937218337141999, 9.570453885520236, 9.833346852000879, 10.39499907358583, 10.73755247843990, 11.32258170939039, 11.57349686313579, 12.20906292779591, 12.71291510720214
