| L(s) = 1 | − 3-s − 3·7-s + 9-s − 5·13-s + 2·17-s + 4·19-s + 3·21-s + 4·23-s − 27-s + 4·29-s + 8·31-s + 7·37-s + 5·39-s − 12·41-s − 5·43-s + 8·47-s + 2·49-s − 2·51-s + 14·53-s − 4·57-s + 4·59-s + 6·61-s − 3·63-s − 12·67-s − 4·69-s − 6·71-s − 5·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.38·13-s + 0.485·17-s + 0.917·19-s + 0.654·21-s + 0.834·23-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 1.15·37-s + 0.800·39-s − 1.87·41-s − 0.762·43-s + 1.16·47-s + 2/7·49-s − 0.280·51-s + 1.92·53-s − 0.529·57-s + 0.520·59-s + 0.768·61-s − 0.377·63-s − 1.46·67-s − 0.481·69-s − 0.712·71-s − 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323871309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323871309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.06121542058401, −14.38139606038772, −13.78570921538781, −13.17694104399637, −12.90758700408535, −12.06173412999995, −11.88413969432993, −11.46804832693610, −10.35198029645724, −10.10250833329100, −9.883685123711790, −9.058169747649808, −8.581316874600501, −7.661994350316103, −7.236320396376837, −6.739865588305488, −6.157152116148664, −5.521834071872854, −4.933942490715081, −4.435249861390367, −3.500261489712690, −2.940042262073210, −2.376719642353200, −1.182311676430812, −0.4943228814487084,
0.4943228814487084, 1.182311676430812, 2.376719642353200, 2.940042262073210, 3.500261489712690, 4.435249861390367, 4.933942490715081, 5.521834071872854, 6.157152116148664, 6.739865588305488, 7.236320396376837, 7.661994350316103, 8.581316874600501, 9.058169747649808, 9.883685123711790, 10.10250833329100, 10.35198029645724, 11.46804832693610, 11.88413969432993, 12.06173412999995, 12.90758700408535, 13.17694104399637, 13.78570921538781, 14.38139606038772, 15.06121542058401
