| L(s) = 1 | + 3-s + 7-s + 9-s − 5·11-s − 13-s + 7·17-s + 6·19-s + 21-s − 3·23-s + 27-s − 2·29-s + 2·31-s − 5·33-s + 7·37-s − 39-s + 9·41-s − 8·43-s − 10·47-s − 6·49-s + 7·51-s + 5·53-s + 6·57-s − 5·61-s + 63-s − 4·67-s − 3·69-s + 9·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s + 0.218·21-s − 0.625·23-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.870·33-s + 1.15·37-s − 0.160·39-s + 1.40·41-s − 1.21·43-s − 1.45·47-s − 6/7·49-s + 0.980·51-s + 0.686·53-s + 0.794·57-s − 0.640·61-s + 0.125·63-s − 0.488·67-s − 0.361·69-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.064354212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.064354212\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.25964599785550, −13.83954612258243, −13.23248625949275, −12.90047627822448, −12.25669317781652, −11.78323948492841, −11.28382270580029, −10.58485819759998, −10.12129831581473, −9.622766681898388, −9.375539885631353, −8.313161629854991, −8.007910901048415, −7.689087952449269, −7.245998109174863, −6.353150925490609, −5.697377906696572, −5.197396857027037, −4.795777586497903, −3.959005173124724, −3.201314341632104, −2.912443787606390, −2.134951746291032, −1.386828418926796, −0.5856872351892453,
0.5856872351892453, 1.386828418926796, 2.134951746291032, 2.912443787606390, 3.201314341632104, 3.959005173124724, 4.795777586497903, 5.197396857027037, 5.697377906696572, 6.353150925490609, 7.245998109174863, 7.689087952449269, 8.007910901048415, 8.313161629854991, 9.375539885631353, 9.622766681898388, 10.12129831581473, 10.58485819759998, 11.28382270580029, 11.78323948492841, 12.25669317781652, 12.90047627822448, 13.23248625949275, 13.83954612258243, 14.25964599785550
