| L(s) = 1 | − 3-s + 3.34·5-s − 2.12·7-s + 9-s + 5.44·11-s − 3.34·15-s − 6.68·17-s + 7.70·19-s + 2.12·21-s + 7.16·23-s + 6.20·25-s − 27-s − 2.79·29-s − 1.30·31-s − 5.44·33-s − 7.12·35-s − 6.38·37-s + 9.98·41-s − 1.95·43-s + 3.34·45-s + 5.32·47-s − 2.46·49-s + 6.68·51-s + 11.2·53-s + 18.2·55-s − 7.70·57-s − 6.53·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49·5-s − 0.804·7-s + 0.333·9-s + 1.64·11-s − 0.864·15-s − 1.62·17-s + 1.76·19-s + 0.464·21-s + 1.49·23-s + 1.24·25-s − 0.192·27-s − 0.518·29-s − 0.234·31-s − 0.947·33-s − 1.20·35-s − 1.04·37-s + 1.55·41-s − 0.298·43-s + 0.498·45-s + 0.776·47-s − 0.352·49-s + 0.935·51-s + 1.54·53-s + 2.45·55-s − 1.02·57-s − 0.850·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.222347398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222347398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 5.44T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 7.80T + 61T^{2} \) |
| 67 | \( 1 - 3.75T + 67T^{2} \) |
| 71 | \( 1 - 3.81T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 + 3.08T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.981439252058305286271771233909, −7.32601208801543790245598404447, −6.79389628665437851013972876152, −6.25452434667037497212855512706, −5.60775823493246607923847119108, −4.85504577390086839777366328965, −3.84918916572871219022892774836, −2.89406283668448176689910283477, −1.82470654782978872064257786362, −0.929161816263477894072568871219,
0.929161816263477894072568871219, 1.82470654782978872064257786362, 2.89406283668448176689910283477, 3.84918916572871219022892774836, 4.85504577390086839777366328965, 5.60775823493246607923847119108, 6.25452434667037497212855512706, 6.79389628665437851013972876152, 7.32601208801543790245598404447, 8.981439252058305286271771233909
