| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·13-s + 15-s + 6·17-s − 4·19-s + 21-s + 23-s + 25-s − 27-s + 6·29-s − 4·31-s + 35-s + 2·37-s − 2·39-s + 10·41-s + 8·43-s − 45-s − 4·47-s + 49-s − 6·51-s − 6·53-s + 4·57-s + 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 1.56·41-s + 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.761901980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761901980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.82147367588858, −14.25829220328163, −13.95585630068333, −12.96425862647965, −12.67860594162375, −12.38848070058482, −11.58925651325730, −11.21018380530100, −10.64589516249493, −10.19942239611291, −9.522488000038834, −9.082623883591179, −8.206593951434711, −7.956513963752150, −7.190435101284849, −6.659534124525762, −6.042910349417934, −5.605363935757790, −4.882878514308038, −4.205916342005080, −3.677605823595776, −2.999544583482584, −2.201266676701108, −1.159951527527469, −0.5854163352028367,
0.5854163352028367, 1.159951527527469, 2.201266676701108, 2.999544583482584, 3.677605823595776, 4.205916342005080, 4.882878514308038, 5.605363935757790, 6.042910349417934, 6.659534124525762, 7.190435101284849, 7.956513963752150, 8.206593951434711, 9.082623883591179, 9.522488000038834, 10.19942239611291, 10.64589516249493, 11.21018380530100, 11.58925651325730, 12.38848070058482, 12.67860594162375, 12.96425862647965, 13.95585630068333, 14.25829220328163, 14.82147367588858
