| L(s) = 1 | + 3-s − 5-s − 0.485·7-s + 9-s − 5.02·11-s + 3.51·13-s − 15-s − 1.32·17-s + 6.64·19-s − 0.485·21-s − 0.292·23-s + 25-s + 27-s − 9.86·29-s − 31-s − 5.02·33-s + 0.485·35-s + 5.51·37-s + 3.51·39-s − 7.02·41-s + 1.02·43-s − 45-s + 6.93·47-s − 6.76·49-s − 1.32·51-s − 1.70·53-s + 5.02·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.183·7-s + 0.333·9-s − 1.51·11-s + 0.974·13-s − 0.258·15-s − 0.320·17-s + 1.52·19-s − 0.106·21-s − 0.0610·23-s + 0.200·25-s + 0.192·27-s − 1.83·29-s − 0.179·31-s − 0.875·33-s + 0.0821·35-s + 0.906·37-s + 0.562·39-s − 1.09·41-s + 0.156·43-s − 0.149·45-s + 1.01·47-s − 0.966·49-s − 0.184·51-s − 0.234·53-s + 0.678·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.485T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 + 0.292T + 23T^{2} \) |
| 29 | \( 1 + 9.86T + 29T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 - 1.02T + 43T^{2} \) |
| 47 | \( 1 - 6.93T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 0.349T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.59426270534025361754137830632, −7.15418654600094475815254049824, −6.07102305949354961583652814087, −5.44685960345031303759820765073, −4.69228577866158452637453018737, −3.73031134491756510545134505540, −3.20666343524690106615393711186, −2.39418084514464639926931843532, −1.32303627365273485511921691362, 0,
1.32303627365273485511921691362, 2.39418084514464639926931843532, 3.20666343524690106615393711186, 3.73031134491756510545134505540, 4.69228577866158452637453018737, 5.44685960345031303759820765073, 6.07102305949354961583652814087, 7.15418654600094475815254049824, 7.59426270534025361754137830632
