| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 3·11-s + 4·13-s − 14-s + 16-s − 8·19-s + 20-s − 3·22-s + 6·23-s − 4·25-s + 4·26-s − 28-s + 9·29-s − 3·31-s + 32-s − 35-s − 6·37-s − 8·38-s + 40-s − 12·41-s + 6·43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.83·19-s + 0.223·20-s − 0.639·22-s + 1.25·23-s − 4/5·25-s + 0.784·26-s − 0.188·28-s + 1.67·29-s − 0.538·31-s + 0.176·32-s − 0.169·35-s − 0.986·37-s − 1.29·38-s + 0.158·40-s − 1.87·41-s + 0.914·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.367605086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.367605086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.94859623217469, −14.31165529891322, −13.67113822391989, −13.32531409423335, −13.01633456804785, −12.34221764748604, −11.96474830271475, −11.04800104938291, −10.66881238138483, −10.43723356572382, −9.606413689629217, −9.015638704316301, −8.241503911118441, −8.118686878278539, −7.000505408215845, −6.565337399131164, −6.225455749519296, −5.394789342649160, −5.026146307399412, −4.270453827242093, −3.582177560394129, −3.034279998072932, −2.254045944152975, −1.685177706023344, −0.5829739267145432,
0.5829739267145432, 1.685177706023344, 2.254045944152975, 3.034279998072932, 3.582177560394129, 4.270453827242093, 5.026146307399412, 5.394789342649160, 6.225455749519296, 6.565337399131164, 7.000505408215845, 8.118686878278539, 8.241503911118441, 9.015638704316301, 9.606413689629217, 10.43723356572382, 10.66881238138483, 11.04800104938291, 11.96474830271475, 12.34221764748604, 13.01633456804785, 13.32531409423335, 13.67113822391989, 14.31165529891322, 14.94859623217469
