| L(s) = 1 | + 2·3-s − 5-s + 9-s + 4·13-s − 2·15-s − 2·19-s + 3·23-s + 25-s − 4·27-s + 9·29-s − 2·31-s + 5·37-s + 8·39-s + 12·41-s + 5·43-s − 45-s − 12·47-s − 9·53-s − 4·57-s + 6·59-s + 10·61-s − 4·65-s − 4·67-s + 6·69-s + 9·71-s + 16·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.769·27-s + 1.67·29-s − 0.359·31-s + 0.821·37-s + 1.28·39-s + 1.87·41-s + 0.762·43-s − 0.149·45-s − 1.75·47-s − 1.23·53-s − 0.529·57-s + 0.781·59-s + 1.28·61-s − 0.496·65-s − 0.488·67-s + 0.722·69-s + 1.06·71-s + 1.87·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.175710252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.175710252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.76004137426074, −12.54013038276563, −11.65737650824720, −11.41727500835634, −10.83430273942267, −10.60520075942819, −9.730849572654949, −9.490408378011940, −8.953953174997184, −8.512209470457055, −8.067512503554162, −7.919830128692605, −7.185828502764859, −6.675686800811655, −6.173463944213957, −5.743599694109950, −4.907923452189608, −4.503903142031186, −3.950570116375279, −3.306284986086615, −3.182959440252816, −2.322253746206066, −2.031612552877109, −1.019620709684995, −0.6554462926324688,
0.6554462926324688, 1.019620709684995, 2.031612552877109, 2.322253746206066, 3.182959440252816, 3.306284986086615, 3.950570116375279, 4.503903142031186, 4.907923452189608, 5.743599694109950, 6.173463944213957, 6.675686800811655, 7.185828502764859, 7.919830128692605, 8.067512503554162, 8.512209470457055, 8.953953174997184, 9.490408378011940, 9.730849572654949, 10.60520075942819, 10.83430273942267, 11.41727500835634, 11.65737650824720, 12.54013038276563, 12.76004137426074
