| L(s) = 1 | + 2-s − 3-s + 4-s − 4.17·5-s − 6-s + 2.87·7-s + 8-s + 9-s − 4.17·10-s − 11-s − 12-s − 3.52·13-s + 2.87·14-s + 4.17·15-s + 16-s − 3.37·17-s + 18-s + 8.24·19-s − 4.17·20-s − 2.87·21-s − 22-s + 0.0460·23-s − 24-s + 12.4·25-s − 3.52·26-s − 27-s + 2.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.86·5-s − 0.408·6-s + 1.08·7-s + 0.353·8-s + 0.333·9-s − 1.32·10-s − 0.301·11-s − 0.288·12-s − 0.977·13-s + 0.767·14-s + 1.07·15-s + 0.250·16-s − 0.817·17-s + 0.235·18-s + 1.89·19-s − 0.934·20-s − 0.626·21-s − 0.213·22-s + 0.00959·23-s − 0.204·24-s + 2.48·25-s − 0.691·26-s − 0.192·27-s + 0.542·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 23 | \( 1 - 0.0460T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 - 1.63T + 47T^{2} \) |
| 53 | \( 1 + 8.16T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 6.24T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.72590191515135165268718797192, −7.45374086656339334548762702658, −6.77525079466163302402819741792, −5.55819221480794169916740021075, −4.81625550579031294061474149282, −4.53848426953079850473838517314, −3.58816325402585388787910826041, −2.74034741034405961075940659534, −1.33769512714066077661193302865, 0,
1.33769512714066077661193302865, 2.74034741034405961075940659534, 3.58816325402585388787910826041, 4.53848426953079850473838517314, 4.81625550579031294061474149282, 5.55819221480794169916740021075, 6.77525079466163302402819741792, 7.45374086656339334548762702658, 7.72590191515135165268718797192
