| L(s) = 1 | − 1.92·2-s + 1.52·3-s + 1.68·4-s − 2.00·5-s − 2.92·6-s + 0.600·8-s − 0.687·9-s + 3.85·10-s − 0.645·11-s + 2.56·12-s − 0.232·13-s − 3.05·15-s − 4.52·16-s + 5.71·17-s + 1.31·18-s − 19-s − 3.38·20-s + 1.23·22-s + 4.21·23-s + 0.913·24-s − 0.972·25-s + 0.447·26-s − 5.60·27-s + 3.65·29-s + 5.86·30-s + 1.75·31-s + 7.49·32-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.878·3-s + 0.843·4-s − 0.897·5-s − 1.19·6-s + 0.212·8-s − 0.229·9-s + 1.21·10-s − 0.194·11-s + 0.740·12-s − 0.0646·13-s − 0.788·15-s − 1.13·16-s + 1.38·17-s + 0.311·18-s − 0.229·19-s − 0.757·20-s + 0.264·22-s + 0.878·23-s + 0.186·24-s − 0.194·25-s + 0.0877·26-s − 1.07·27-s + 0.678·29-s + 1.07·30-s + 0.314·31-s + 1.32·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8093519062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8093519062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 + 2.00T + 5T^{2} \) |
| 11 | \( 1 + 0.645T + 11T^{2} \) |
| 13 | \( 1 + 0.232T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 0.719T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 + 6.14T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−9.856595024733387545213261895645, −9.038034100348826855447477538417, −8.486316156566803928007660275146, −7.65154926045590901063365881681, −7.42379045578393621963517110378, −5.94597818618384745741944477060, −4.57058161335807970360234894860, −3.45864166043724270580904270322, −2.42024942320379549859341272355, −0.840800211282180458626110222949,
0.840800211282180458626110222949, 2.42024942320379549859341272355, 3.45864166043724270580904270322, 4.57058161335807970360234894860, 5.94597818618384745741944477060, 7.42379045578393621963517110378, 7.65154926045590901063365881681, 8.486316156566803928007660275146, 9.038034100348826855447477538417, 9.856595024733387545213261895645
