| L(s) = 1 | + 3.03·3-s + 5-s + 3.09·7-s + 6.18·9-s + 5.41·11-s − 1.41·13-s + 3.03·15-s − 1.72·17-s − 2.75·19-s + 9.37·21-s + 1.19·23-s + 25-s + 9.65·27-s − 4.80·29-s + 1.78·31-s + 16.4·33-s + 3.09·35-s + 0.781·37-s − 4.29·39-s + 3.31·41-s − 6.64·43-s + 6.18·45-s + 1.28·47-s + 2.57·49-s − 5.22·51-s + 2.00·53-s + 5.41·55-s + ⋯ |
| L(s) = 1 | + 1.74·3-s + 0.447·5-s + 1.16·7-s + 2.06·9-s + 1.63·11-s − 0.393·13-s + 0.782·15-s − 0.418·17-s − 0.630·19-s + 2.04·21-s + 0.250·23-s + 0.200·25-s + 1.85·27-s − 0.892·29-s + 0.320·31-s + 2.85·33-s + 0.523·35-s + 0.128·37-s − 0.687·39-s + 0.517·41-s − 1.01·43-s + 0.922·45-s + 0.187·47-s + 0.368·49-s − 0.731·51-s + 0.275·53-s + 0.730·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.527932932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.527932932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 3.03T + 3T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 + 6.64T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 0.682T + 79T^{2} \) |
| 83 | \( 1 - 0.300T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−8.363221025262248844620797409989, −7.43914612654378120912398682381, −6.93025246509342000809893891169, −6.07421553940072915850341431158, −4.92990693654226220899273053209, −4.25005395754095190798798681812, −3.66841685657023754368874512752, −2.65052324159717049782406770004, −1.89802551420748863281970268609, −1.33450514968962826098361401548,
1.33450514968962826098361401548, 1.89802551420748863281970268609, 2.65052324159717049782406770004, 3.66841685657023754368874512752, 4.25005395754095190798798681812, 4.92990693654226220899273053209, 6.07421553940072915850341431158, 6.93025246509342000809893891169, 7.43914612654378120912398682381, 8.363221025262248844620797409989
