| L(s) = 1 | + 2.66i·3-s − 3.66i·7-s − 4.12·9-s − 1.21·11-s + 2.21i·13-s − 1.21i·17-s + 2.57·19-s + 9.79·21-s + i·23-s − 3.00i·27-s − 1.45·29-s − 6.46·31-s − 3.24i·33-s − 4i·37-s − 5.90·39-s + ⋯ |
| L(s) = 1 | + 1.54i·3-s − 1.38i·7-s − 1.37·9-s − 0.366·11-s + 0.614i·13-s − 0.294i·17-s + 0.591·19-s + 2.13·21-s + 0.208i·23-s − 0.577i·27-s − 0.270·29-s − 1.16·31-s − 0.564i·33-s − 0.657i·37-s − 0.946·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8053561500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8053561500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 - 2.66iT - 3T^{2} \) |
| 7 | \( 1 + 3.66iT - 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 - 2.21iT - 13T^{2} \) |
| 17 | \( 1 + 1.21iT - 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 6.90iT - 43T^{2} \) |
| 47 | \( 1 + 5.45iT - 47T^{2} \) |
| 53 | \( 1 - 3.81iT - 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 + 6.91T + 71T^{2} \) |
| 73 | \( 1 + 15.2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 1.69iT - 97T^{2} \) |
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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
−8.263698512240336105506373485944, −7.43176737154185137243283328064, −6.87568316066348355385048410985, −5.77602099185594838681160732704, −4.99330874192870947658213427225, −4.44514158840672353155207747066, −3.68186825769584694207014936663, −3.19504905852328716046422497621, −1.73696861946420270498146359549, −0.22755130811372979630780094344,
1.15457487375190425493734328003, 2.13165625011420167762150851159, 2.69490317863187618736973825999, 3.67604046992134959270165361614, 5.24373888982534680842423255941, 5.50475642251607866870889554286, 6.35437238795840323404033792350, 7.01369410062183134363355799737, 7.71427249953301410615493773767, 8.401774413761988900295450274876
