| L(s) = 1 | + 1.01·2-s − 3-s − 0.961·4-s + 2.85·5-s − 1.01·6-s − 3.01·8-s + 9-s + 2.90·10-s + 1.04·11-s + 0.961·12-s − 4.67·13-s − 2.85·15-s − 1.15·16-s + 1.79·17-s + 1.01·18-s − 2.26·19-s − 2.74·20-s + 1.06·22-s + 23-s + 3.01·24-s + 3.15·25-s − 4.76·26-s − 27-s + 3.86·29-s − 2.90·30-s + 9.09·31-s + 4.86·32-s + ⋯ |
| L(s) = 1 | + 0.720·2-s − 0.577·3-s − 0.480·4-s + 1.27·5-s − 0.416·6-s − 1.06·8-s + 0.333·9-s + 0.920·10-s + 0.314·11-s + 0.277·12-s − 1.29·13-s − 0.737·15-s − 0.288·16-s + 0.434·17-s + 0.240·18-s − 0.519·19-s − 0.613·20-s + 0.226·22-s + 0.208·23-s + 0.616·24-s + 0.630·25-s − 0.935·26-s − 0.192·27-s + 0.717·29-s − 0.531·30-s + 1.63·31-s + 0.859·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148729817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148729817\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.01T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 41 | \( 1 - 7.76T + 41T^{2} \) |
| 43 | \( 1 + 0.798T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 6.50T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 + 7.52T + 67T^{2} \) |
| 71 | \( 1 + 0.379T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.809003215957157702315719153893, −7.79400244437991582086392791320, −6.78414788192226182761500632092, −6.11780579330888384097459123397, −5.57857087272567569053211970343, −4.80963944979972364623492762581, −4.29071759133285087785979630808, −3.01619940495465140523308130525, −2.19652151886472910695673480891, −0.810315865276667288594431625511,
0.810315865276667288594431625511, 2.19652151886472910695673480891, 3.01619940495465140523308130525, 4.29071759133285087785979630808, 4.80963944979972364623492762581, 5.57857087272567569053211970343, 6.11780579330888384097459123397, 6.78414788192226182761500632092, 7.79400244437991582086392791320, 8.809003215957157702315719153893
