| L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 2·12-s + 4·16-s + 6·17-s + 19-s + 21-s + 6·23-s − 5·25-s + 27-s − 2·28-s + 7·31-s − 2·36-s − 5·37-s − 4·43-s + 6·47-s + 4·48-s − 6·49-s + 6·51-s + 57-s + 6·59-s − 61-s + 63-s − 8·64-s − 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 16-s + 1.45·17-s + 0.229·19-s + 0.218·21-s + 1.25·23-s − 25-s + 0.192·27-s − 0.377·28-s + 1.25·31-s − 1/3·36-s − 0.821·37-s − 0.609·43-s + 0.875·47-s + 0.577·48-s − 6/7·49-s + 0.840·51-s + 0.132·57-s + 0.781·59-s − 0.128·61-s + 0.125·63-s − 64-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.49569278663028, −14.02963113978549, −13.60115487423669, −13.14878304303152, −12.63112792711829, −12.02487666688952, −11.68047086871850, −10.91595629493908, −10.17202650378304, −9.972991684649857, −9.431314565554305, −8.778387703852766, −8.429430861427638, −7.906900061591647, −7.404295592084341, −6.856127331403618, −5.892525573762358, −5.527327015136489, −4.851900550674176, −4.402780363602902, −3.682344429622198, −3.192487033145982, −2.592806955488869, −1.466856810460606, −1.110780730055684, 0,
1.110780730055684, 1.466856810460606, 2.592806955488869, 3.192487033145982, 3.682344429622198, 4.402780363602902, 4.851900550674176, 5.527327015136489, 5.892525573762358, 6.856127331403618, 7.404295592084341, 7.906900061591647, 8.429430861427638, 8.778387703852766, 9.431314565554305, 9.972991684649857, 10.17202650378304, 10.91595629493908, 11.68047086871850, 12.02487666688952, 12.63112792711829, 13.14878304303152, 13.60115487423669, 14.02963113978549, 14.49569278663028
