| L(s) = 1 | + 2.08·2-s + 2.34·4-s − 2.98·5-s + 3.78·7-s + 0.718·8-s − 6.21·10-s − 11-s + 4.41·13-s + 7.89·14-s − 3.19·16-s − 5.30·17-s + 0.311·19-s − 6.99·20-s − 2.08·22-s + 5.62·23-s + 3.89·25-s + 9.20·26-s + 8.87·28-s + 8.30·29-s − 6.85·31-s − 8.08·32-s − 11.0·34-s − 11.2·35-s + 4.98·37-s + 0.650·38-s − 2.14·40-s + 11.9·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s − 1.33·5-s + 1.43·7-s + 0.254·8-s − 1.96·10-s − 0.301·11-s + 1.22·13-s + 2.10·14-s − 0.797·16-s − 1.28·17-s + 0.0715·19-s − 1.56·20-s − 0.444·22-s + 1.17·23-s + 0.778·25-s + 1.80·26-s + 1.67·28-s + 1.54·29-s − 1.23·31-s − 1.43·32-s − 1.89·34-s − 1.90·35-s + 0.818·37-s + 0.105·38-s − 0.338·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.156092428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.156092428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 - 0.311T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 4.98T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 0.135T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 - 7.08T + 59T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.87095515119508690721849364933, −7.35156878173500650820405529924, −6.50747716833542214884618880728, −5.75942698226128389146850887942, −4.88449827420827901106657718539, −4.44334940000189265520654722996, −3.93531859912684369571560503990, −3.07933218218313319372822350067, −2.18078082390003323212869092470, −0.872999110720427619735919697770,
0.872999110720427619735919697770, 2.18078082390003323212869092470, 3.07933218218313319372822350067, 3.93531859912684369571560503990, 4.44334940000189265520654722996, 4.88449827420827901106657718539, 5.75942698226128389146850887942, 6.50747716833542214884618880728, 7.35156878173500650820405529924, 7.87095515119508690721849364933
