L(s) = 1 | − 2.45·2-s + 4.02·4-s − 1.44·5-s + 0.736·7-s − 4.96·8-s + 3.53·10-s + 0.631·11-s + 0.659·13-s − 1.80·14-s + 4.12·16-s − 3.60·17-s + 3.07·19-s − 5.80·20-s − 1.54·22-s − 7.24·23-s − 2.91·25-s − 1.61·26-s + 2.96·28-s − 4.33·29-s − 4.66·31-s − 0.211·32-s + 8.84·34-s − 1.06·35-s + 8.87·37-s − 7.53·38-s + 7.15·40-s + 8.77·41-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.01·4-s − 0.645·5-s + 0.278·7-s − 1.75·8-s + 1.11·10-s + 0.190·11-s + 0.182·13-s − 0.483·14-s + 1.03·16-s − 0.873·17-s + 0.704·19-s − 1.29·20-s − 0.330·22-s − 1.51·23-s − 0.583·25-s − 0.317·26-s + 0.559·28-s − 0.804·29-s − 0.837·31-s − 0.0373·32-s + 1.51·34-s − 0.179·35-s + 1.45·37-s − 1.22·38-s + 1.13·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5126415103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5126415103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 0.736T + 7T^{2} \) |
| 11 | \( 1 - 0.631T + 11T^{2} \) |
| 13 | \( 1 - 0.659T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 4.33T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 + 5.16T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.0719T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 7.19T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−8.320758410767123408449942849771, −7.66355441555976740913283858022, −7.40120111782500387740464936966, −6.38170403670294814554017086216, −5.74602783151353983232906591230, −4.44992870028904839784912218029, −3.69063093490014148285044844572, −2.46102691347313846462848935994, −1.68734117089265386422044918061, −0.51803066222371355605115655614,
0.51803066222371355605115655614, 1.68734117089265386422044918061, 2.46102691347313846462848935994, 3.69063093490014148285044844572, 4.44992870028904839784912218029, 5.74602783151353983232906591230, 6.38170403670294814554017086216, 7.40120111782500387740464936966, 7.66355441555976740913283858022, 8.320758410767123408449942849771