L(s) = 1 | + 3-s − 2·4-s − 2.55·5-s + 9-s + 6.17·11-s − 2·12-s + 2.06·13-s − 2.55·15-s + 4·16-s + 2·17-s + 0.447·19-s + 5.10·20-s − 0.552·23-s + 1.51·25-s + 27-s − 5.10·29-s + 7.72·31-s + 6.17·33-s − 2·36-s − 37-s + 2.06·39-s + 41-s − 10.6·43-s − 12.3·44-s − 2.55·45-s + 8.30·47-s + 4·48-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 1.14·5-s + 0.333·9-s + 1.86·11-s − 0.577·12-s + 0.573·13-s − 0.659·15-s + 16-s + 0.485·17-s + 0.102·19-s + 1.14·20-s − 0.115·23-s + 0.303·25-s + 0.192·27-s − 0.947·29-s + 1.38·31-s + 1.07·33-s − 0.333·36-s − 0.164·37-s + 0.331·39-s + 0.156·41-s − 1.61·43-s − 1.86·44-s − 0.380·45-s + 1.21·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.732730885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.732730885\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 11 | \( 1 - 6.17T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.447T + 19T^{2} \) |
| 23 | \( 1 + 0.552T + 23T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 - 7.72T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 8.30T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 + 7.75T + 61T^{2} \) |
| 67 | \( 1 + 0.0676T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.127641987375959973680457348880, −7.63386547346135961374643445937, −6.71694844932871923206438935852, −6.02984364808429978834508874751, −4.93435940596202892189719674817, −4.20205410891521427056365964856, −3.72032357626640327382059605224, −3.22286313169863185415715864309, −1.63559194624772341383681148008, −0.72624118072804194287913613433,
0.72624118072804194287913613433, 1.63559194624772341383681148008, 3.22286313169863185415715864309, 3.72032357626640327382059605224, 4.20205410891521427056365964856, 4.93435940596202892189719674817, 6.02984364808429978834508874751, 6.71694844932871923206438935852, 7.63386547346135961374643445937, 8.127641987375959973680457348880