| L(s) = 1 | + 1.95·2-s + 3-s + 1.80·4-s + 2.46·5-s + 1.95·6-s − 0.380·8-s + 9-s + 4.81·10-s + 3.40·11-s + 1.80·12-s + 3.78·13-s + 2.46·15-s − 4.35·16-s − 7.48·17-s + 1.95·18-s + 1.93·19-s + 4.45·20-s + 6.65·22-s − 1.38·23-s − 0.380·24-s + 1.08·25-s + 7.38·26-s + 27-s + 4.33·29-s + 4.81·30-s + 10.1·31-s − 7.72·32-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.577·3-s + 0.902·4-s + 1.10·5-s + 0.796·6-s − 0.134·8-s + 0.333·9-s + 1.52·10-s + 1.02·11-s + 0.521·12-s + 1.05·13-s + 0.636·15-s − 1.08·16-s − 1.81·17-s + 0.459·18-s + 0.443·19-s + 0.995·20-s + 1.41·22-s − 0.289·23-s − 0.0776·24-s + 0.217·25-s + 1.44·26-s + 0.192·27-s + 0.804·29-s + 0.878·30-s + 1.82·31-s − 1.36·32-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
| \(L(1)\) |
\(\approx\) |
\(6.967321305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.967321305\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.95T + 2T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 7.48T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.77T + 37T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 4.49T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 0.523T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−8.241372927573300143432502069456, −6.87389028646082321829985140312, −6.44311903196690567031409838177, −6.02942106636778111493866466574, −5.10194350320206135745944740879, −4.29927307075122516600365877994, −3.86910518091299418785053535976, −2.80987556764876197190681651284, −2.25187325509826837477033862804, −1.19687712711876253147484949016,
1.19687712711876253147484949016, 2.25187325509826837477033862804, 2.80987556764876197190681651284, 3.86910518091299418785053535976, 4.29927307075122516600365877994, 5.10194350320206135745944740879, 6.02942106636778111493866466574, 6.44311903196690567031409838177, 6.87389028646082321829985140312, 8.241372927573300143432502069456
