| L(s) = 1 | + 2-s − 2.09·3-s + 4-s + 3.98·5-s − 2.09·6-s + 1.58·7-s + 8-s + 1.40·9-s + 3.98·10-s + 3.27·11-s − 2.09·12-s + 2.09·13-s + 1.58·14-s − 8.36·15-s + 16-s + 3.89·17-s + 1.40·18-s − 2.88·19-s + 3.98·20-s − 3.32·21-s + 3.27·22-s + 5.91·23-s − 2.09·24-s + 10.8·25-s + 2.09·26-s + 3.35·27-s + 1.58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.21·3-s + 0.5·4-s + 1.78·5-s − 0.856·6-s + 0.598·7-s + 0.353·8-s + 0.467·9-s + 1.26·10-s + 0.988·11-s − 0.605·12-s + 0.581·13-s + 0.423·14-s − 2.15·15-s + 0.250·16-s + 0.944·17-s + 0.330·18-s − 0.662·19-s + 0.891·20-s − 0.724·21-s + 0.699·22-s + 1.23·23-s − 0.428·24-s + 2.17·25-s + 0.410·26-s + 0.645·27-s + 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.806796457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806796457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 2.09T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 5.91T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 - 0.0545T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 - 0.311T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 + 4.74T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 4.23T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
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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
−7.958849112087665547441201411459, −6.80118168321111315897508508574, −6.52315618235789273271568238369, −5.72770611064611842710448666234, −5.43126963589504055111243817986, −4.78217523831583211305220021361, −3.80406174473188906467542140479, −2.73223584931604799197703431797, −1.67381100296313213456743758188, −1.11532802929446960491750395199,
1.11532802929446960491750395199, 1.67381100296313213456743758188, 2.73223584931604799197703431797, 3.80406174473188906467542140479, 4.78217523831583211305220021361, 5.43126963589504055111243817986, 5.72770611064611842710448666234, 6.52315618235789273271568238369, 6.80118168321111315897508508574, 7.958849112087665547441201411459
