| L(s) = 1 | + 2-s + 2.15·3-s + 4-s − 5-s + 2.15·6-s + 5.20·7-s + 8-s + 1.64·9-s − 10-s + 11-s + 2.15·12-s − 3.69·13-s + 5.20·14-s − 2.15·15-s + 16-s + 2.28·17-s + 1.64·18-s + 3.91·19-s − 20-s + 11.2·21-s + 22-s − 7.46·23-s + 2.15·24-s + 25-s − 3.69·26-s − 2.92·27-s + 5.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.24·3-s + 0.5·4-s − 0.447·5-s + 0.879·6-s + 1.96·7-s + 0.353·8-s + 0.547·9-s − 0.316·10-s + 0.301·11-s + 0.621·12-s − 1.02·13-s + 1.38·14-s − 0.556·15-s + 0.250·16-s + 0.555·17-s + 0.386·18-s + 0.897·19-s − 0.223·20-s + 2.44·21-s + 0.213·22-s − 1.55·23-s + 0.439·24-s + 0.200·25-s − 0.724·26-s − 0.563·27-s + 0.982·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.729274703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.729274703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 2.15T + 3T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 + 0.670T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 4.29T + 67T^{2} \) |
| 71 | \( 1 + 8.54T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 6.18T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 - 5.40T + 97T^{2} \) |
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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
−8.003824207051334077188994020734, −7.81841627456593421124539964541, −7.16488784576620374672980352314, −5.94526286761694397523269975617, −5.11857666127084774466248714741, −4.46452603020322755326199427041, −3.86264456965541670131384762016, −2.84260439365552681805021233944, −2.18773237801882279681514706146, −1.25686867615493973001189647160,
1.25686867615493973001189647160, 2.18773237801882279681514706146, 2.84260439365552681805021233944, 3.86264456965541670131384762016, 4.46452603020322755326199427041, 5.11857666127084774466248714741, 5.94526286761694397523269975617, 7.16488784576620374672980352314, 7.81841627456593421124539964541, 8.003824207051334077188994020734
