| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3.12·7-s + 8-s + 9-s + 10-s − 3.12·11-s − 12-s + 2·13-s + 3.12·14-s − 15-s + 16-s − 1.12·17-s + 18-s + 4·19-s + 20-s − 3.12·21-s − 3.12·22-s + 23-s − 24-s + 25-s + 2·26-s − 27-s + 3.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.941·11-s − 0.288·12-s + 0.554·13-s + 0.834·14-s − 0.258·15-s + 0.250·16-s − 0.272·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.681·21-s − 0.665·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.331129555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.331129555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−10.70347967111563947986137354215, −9.893093454040967770924047872289, −8.600193152708372041192707561691, −7.72790984760758369525384065297, −6.80283966174642378824177880289, −5.66120849697089272939956247000, −5.16330293180140880866797513771, −4.21035356013394717146926553200, −2.73794016074583526324836268727, −1.41641795769549472108387942963,
1.41641795769549472108387942963, 2.73794016074583526324836268727, 4.21035356013394717146926553200, 5.16330293180140880866797513771, 5.66120849697089272939956247000, 6.80283966174642378824177880289, 7.72790984760758369525384065297, 8.600193152708372041192707561691, 9.893093454040967770924047872289, 10.70347967111563947986137354215
