| L(s) = 1 | + 2.62·2-s − 3-s + 4.90·4-s + 1.82·5-s − 2.62·6-s + 7.64·8-s + 9-s + 4.80·10-s + 3.28·11-s − 4.90·12-s + 0.121·13-s − 1.82·15-s + 10.2·16-s + 1.01·17-s + 2.62·18-s − 0.491·19-s + 8.97·20-s + 8.64·22-s + 5.54·23-s − 7.64·24-s − 1.65·25-s + 0.318·26-s − 27-s + 1.44·29-s − 4.80·30-s − 7.25·31-s + 11.7·32-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.45·4-s + 0.817·5-s − 1.07·6-s + 2.70·8-s + 0.333·9-s + 1.51·10-s + 0.991·11-s − 1.41·12-s + 0.0336·13-s − 0.472·15-s + 2.56·16-s + 0.246·17-s + 0.619·18-s − 0.112·19-s + 2.00·20-s + 1.84·22-s + 1.15·23-s − 1.56·24-s − 0.331·25-s + 0.0625·26-s − 0.192·27-s + 0.267·29-s − 0.877·30-s − 1.30·31-s + 2.07·32-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
| \(L(1)\) |
\(\approx\) |
\(7.466519352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.466519352\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.62T + 2T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 - 3.28T + 11T^{2} \) |
| 13 | \( 1 - 0.121T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 + 0.491T + 19T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 + 7.25T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 2.81T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 + 7.10T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 2.33T + 97T^{2} \) |
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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
−7.54145334778852067892792452526, −7.02126977106806357364414000778, −6.33995573338634169141274114518, −5.76390490132473195079355478870, −5.32861575375241115004092663649, −4.46629517336143278708613252377, −3.86750387133518746921816796582, −3.00553871765565921617804368897, −2.07073665532992661701611452593, −1.24703787848965253497029029804,
1.24703787848965253497029029804, 2.07073665532992661701611452593, 3.00553871765565921617804368897, 3.86750387133518746921816796582, 4.46629517336143278708613252377, 5.32861575375241115004092663649, 5.76390490132473195079355478870, 6.33995573338634169141274114518, 7.02126977106806357364414000778, 7.54145334778852067892792452526
