| L(s) = 1 | − 3-s + 2.44·5-s − 7-s + 9-s + 0.449·11-s + 6.89·13-s − 2.44·15-s + 1.55·17-s + 6.89·19-s + 21-s + 4.44·23-s + 0.999·25-s − 27-s + 6.89·29-s + 2.89·31-s − 0.449·33-s − 2.44·35-s − 2·37-s − 6.89·39-s + 7.34·41-s − 12.8·43-s + 2.44·45-s − 8.89·47-s + 49-s − 1.55·51-s − 10·53-s + 1.10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.09·5-s − 0.377·7-s + 0.333·9-s + 0.135·11-s + 1.91·13-s − 0.632·15-s + 0.376·17-s + 1.58·19-s + 0.218·21-s + 0.927·23-s + 0.199·25-s − 0.192·27-s + 1.28·29-s + 0.520·31-s − 0.0782·33-s − 0.414·35-s − 0.328·37-s − 1.10·39-s + 1.14·41-s − 1.96·43-s + 0.365·45-s − 1.29·47-s + 0.142·49-s − 0.217·51-s − 1.37·53-s + 0.148·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466446556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466446556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 0.449T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 - 4.44T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 79 | \( 1 + 3.10T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.303884433830111861966787166098, −7.30024699782449396220593784841, −6.45973483613623277103401740760, −6.12865365807960846870641605159, −5.39965968314420612940774502107, −4.72790362872200368450615652547, −3.54280303251370920413461316352, −2.96731371383366497259352898662, −1.57903243034526033198796369456, −0.982644963701242080047788468198,
0.982644963701242080047788468198, 1.57903243034526033198796369456, 2.96731371383366497259352898662, 3.54280303251370920413461316352, 4.72790362872200368450615652547, 5.39965968314420612940774502107, 6.12865365807960846870641605159, 6.45973483613623277103401740760, 7.30024699782449396220593784841, 8.303884433830111861966787166098
