L(s) = 1 | − 1.74·2-s + 1.04·4-s − 3.84·5-s + 0.0845·7-s + 1.66·8-s + 6.71·10-s − 0.570·11-s + 1.00·13-s − 0.147·14-s − 4.99·16-s + 6.52·17-s + 0.290·19-s − 4.03·20-s + 0.995·22-s − 1.26·23-s + 9.78·25-s − 1.75·26-s + 0.0886·28-s − 8.61·29-s + 4.67·31-s + 5.40·32-s − 11.3·34-s − 0.324·35-s + 8.22·37-s − 0.506·38-s − 6.38·40-s + 1.13·41-s + ⋯ |
L(s) = 1 | − 1.23·2-s + 0.524·4-s − 1.71·5-s + 0.0319·7-s + 0.587·8-s + 2.12·10-s − 0.171·11-s + 0.279·13-s − 0.0394·14-s − 1.24·16-s + 1.58·17-s + 0.0665·19-s − 0.901·20-s + 0.212·22-s − 0.264·23-s + 1.95·25-s − 0.345·26-s + 0.0167·28-s − 1.60·29-s + 0.839·31-s + 0.955·32-s − 1.95·34-s − 0.0549·35-s + 1.35·37-s − 0.0822·38-s − 1.00·40-s + 0.176·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5378083693\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5378083693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 - 0.0845T + 7T^{2} \) |
| 11 | \( 1 + 0.570T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 - 0.290T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 - 8.22T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 7.64T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 0.577T + 59T^{2} \) |
| 61 | \( 1 + 0.979T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.27T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.781T + 97T^{2} \) |
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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
−8.134441018982792506131787217925, −7.55956677093669464972514398883, −7.21608954520465927129735920852, −6.11658448440575405540171296704, −5.14363940431082247581674858315, −4.25578401291994647378930230480, −3.70114049550539897313946832844, −2.75418894948970465659976330969, −1.37079016886016221972061157032, −0.51601359882115087752248430348,
0.51601359882115087752248430348, 1.37079016886016221972061157032, 2.75418894948970465659976330969, 3.70114049550539897313946832844, 4.25578401291994647378930230480, 5.14363940431082247581674858315, 6.11658448440575405540171296704, 7.21608954520465927129735920852, 7.55956677093669464972514398883, 8.134441018982792506131787217925