| L(s) = 1 | + 3-s − 5-s + 1.05·7-s + 9-s + 0.701·11-s − 5.99·13-s − 15-s − 4.59·17-s + 2.17·19-s + 1.05·21-s + 3.01·23-s + 25-s + 27-s + 6.98·29-s + 6.84·31-s + 0.701·33-s − 1.05·35-s − 4.03·37-s − 5.99·39-s + 2.54·41-s + 3.22·43-s − 45-s + 9.92·47-s − 5.89·49-s − 4.59·51-s − 11.6·53-s − 0.701·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.397·7-s + 0.333·9-s + 0.211·11-s − 1.66·13-s − 0.258·15-s − 1.11·17-s + 0.498·19-s + 0.229·21-s + 0.629·23-s + 0.200·25-s + 0.192·27-s + 1.29·29-s + 1.22·31-s + 0.122·33-s − 0.177·35-s − 0.663·37-s − 0.959·39-s + 0.397·41-s + 0.492·43-s − 0.149·45-s + 1.44·47-s − 0.841·49-s − 0.642·51-s − 1.59·53-s − 0.0945·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.089268196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089268196\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.701T + 11T^{2} \) |
| 13 | \( 1 + 5.99T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 - 6.84T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 71 | \( 1 - 8.35T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 4.35T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.299549860136568706838271138070, −7.85826945612538498235851192620, −6.98556092560985821251581818927, −6.55389180243910808399147436100, −5.14011291606597331127156641100, −4.72358089480483244601612294131, −3.87837823957319339731846063190, −2.80678235765138250340047424028, −2.20630203692162372933231207013, −0.799204878053610257137096012836,
0.799204878053610257137096012836, 2.20630203692162372933231207013, 2.80678235765138250340047424028, 3.87837823957319339731846063190, 4.72358089480483244601612294131, 5.14011291606597331127156641100, 6.55389180243910808399147436100, 6.98556092560985821251581818927, 7.85826945612538498235851192620, 8.299549860136568706838271138070
