| L(s) = 1 | − 3.11·3-s − 2.45·5-s − 1.48·7-s + 6.68·9-s + 4.66·11-s + 4.68·13-s + 7.64·15-s + 6.87·17-s − 2.20·19-s + 4.62·21-s + 1.02·25-s − 11.4·27-s + 5.97·29-s + 4.71·31-s − 14.5·33-s + 3.64·35-s − 7.60·37-s − 14.5·39-s + 4.25·41-s + 0.0505·43-s − 16.4·45-s + 8.36·47-s − 4.78·49-s − 21.4·51-s + 7.16·53-s − 11.4·55-s + 6.87·57-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 1.09·5-s − 0.561·7-s + 2.22·9-s + 1.40·11-s + 1.30·13-s + 1.97·15-s + 1.66·17-s − 0.506·19-s + 1.00·21-s + 0.205·25-s − 2.20·27-s + 1.10·29-s + 0.846·31-s − 2.52·33-s + 0.616·35-s − 1.24·37-s − 2.33·39-s + 0.664·41-s + 0.00770·43-s − 2.44·45-s + 1.22·47-s − 0.684·49-s − 2.99·51-s + 0.984·53-s − 1.54·55-s + 0.910·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8945370129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8945370129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 29 | \( 1 - 5.97T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 0.0505T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 - 7.16T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 1.96T + 61T^{2} \) |
| 67 | \( 1 - 0.723T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 9.33T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 - 4.19T + 89T^{2} \) |
| 97 | \( 1 + 6.12T + 97T^{2} \) |
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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
−8.303145943547830345670794297954, −7.47346929320740794625167921569, −6.68933829609360810234331462757, −6.23065935742508697227752564089, −5.62973129545562341774097373276, −4.60129136237140604846501826025, −3.95182321504162576251872258769, −3.34762583808882262200015664336, −1.35469701058964644050211838870, −0.68052083036249587872546033615,
0.68052083036249587872546033615, 1.35469701058964644050211838870, 3.34762583808882262200015664336, 3.95182321504162576251872258769, 4.60129136237140604846501826025, 5.62973129545562341774097373276, 6.23065935742508697227752564089, 6.68933829609360810234331462757, 7.47346929320740794625167921569, 8.303145943547830345670794297954
