| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.56·7-s + 8-s + 9-s + 10-s − 4.12·11-s + 12-s + 3.56·14-s + 15-s + 16-s + 5.12·17-s + 18-s − 3.56·19-s + 20-s + 3.56·21-s − 4.12·22-s − 7.68·23-s + 24-s + 25-s + 27-s + 3.56·28-s + 6.56·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.34·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.24·11-s + 0.288·12-s + 0.951·14-s + 0.258·15-s + 0.250·16-s + 1.24·17-s + 0.235·18-s − 0.817·19-s + 0.223·20-s + 0.777·21-s − 0.879·22-s − 1.60·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.673·28-s + 1.21·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.987572579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.987572579\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 4.87T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 7.43T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.104496163096485352392188819357, −7.73110271195495359051022879707, −6.76789969932272338760958370498, −5.83283435058607365525547919130, −5.28587988422669838596121846405, −4.54008934964114189254564718901, −3.85675563620953944982759087698, −2.64923246643240239205773349305, −2.23935305887453758019212679292, −1.14368476076863984087034754832,
1.14368476076863984087034754832, 2.23935305887453758019212679292, 2.64923246643240239205773349305, 3.85675563620953944982759087698, 4.54008934964114189254564718901, 5.28587988422669838596121846405, 5.83283435058607365525547919130, 6.76789969932272338760958370498, 7.73110271195495359051022879707, 8.104496163096485352392188819357
