| L(s) = 1 | + 2-s − 1.16·3-s + 4-s − 5-s − 1.16·6-s − 3.85·7-s + 8-s − 1.63·9-s − 10-s + 11-s − 1.16·12-s − 3.76·13-s − 3.85·14-s + 1.16·15-s + 16-s − 5.05·17-s − 1.63·18-s − 2.28·19-s − 20-s + 4.50·21-s + 22-s − 3.15·23-s − 1.16·24-s + 25-s − 3.76·26-s + 5.41·27-s − 3.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.673·3-s + 0.5·4-s − 0.447·5-s − 0.476·6-s − 1.45·7-s + 0.353·8-s − 0.546·9-s − 0.316·10-s + 0.301·11-s − 0.336·12-s − 1.04·13-s − 1.03·14-s + 0.301·15-s + 0.250·16-s − 1.22·17-s − 0.386·18-s − 0.525·19-s − 0.223·20-s + 0.982·21-s + 0.213·22-s − 0.658·23-s − 0.238·24-s + 0.200·25-s − 0.738·26-s + 1.04·27-s − 0.729·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8087171664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8087171664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.16T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 9.20T + 41T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 + 0.268T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.390966813611691551582487356360, −7.02024296772372066633221579842, −6.86009992666465857793024361351, −6.11397653782986215230030194932, −5.39982161220513748027402385141, −4.57729339285734707995611765627, −3.85088013505766089229150550556, −2.97132858954179964377092749564, −2.24186852053164280242766350630, −0.42768836711532464038692326250,
0.42768836711532464038692326250, 2.24186852053164280242766350630, 2.97132858954179964377092749564, 3.85088013505766089229150550556, 4.57729339285734707995611765627, 5.39982161220513748027402385141, 6.11397653782986215230030194932, 6.86009992666465857793024361351, 7.02024296772372066633221579842, 8.390966813611691551582487356360
