| L(s) = 1 | − 2.37·2-s − 3-s + 3.62·4-s − 1.47·5-s + 2.37·6-s − 3.84·8-s + 9-s + 3.49·10-s + 0.875·11-s − 3.62·12-s − 6.74·13-s + 1.47·15-s + 1.87·16-s + 6.71·17-s − 2.37·18-s − 19-s − 5.34·20-s − 2.07·22-s − 0.875·23-s + 3.84·24-s − 2.82·25-s + 15.9·26-s − 27-s + 0.599·29-s − 3.49·30-s − 2.87·31-s + 3.24·32-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.81·4-s − 0.659·5-s + 0.967·6-s − 1.35·8-s + 0.333·9-s + 1.10·10-s + 0.263·11-s − 1.04·12-s − 1.86·13-s + 0.380·15-s + 0.468·16-s + 1.62·17-s − 0.558·18-s − 0.229·19-s − 1.19·20-s − 0.442·22-s − 0.182·23-s + 0.785·24-s − 0.564·25-s + 3.13·26-s − 0.192·27-s + 0.111·29-s − 0.638·30-s − 0.516·31-s + 0.573·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3293309986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3293309986\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 - 0.875T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 23 | \( 1 + 0.875T + 23T^{2} \) |
| 29 | \( 1 - 0.599T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 0.294T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 4.08T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.855690596695143811938851766793, −7.948322873934812182251678547628, −7.44910989861900661481912566546, −7.01541488468522217454750251038, −5.92630099870698375050312139866, −5.05525167415767130788780065649, −3.99853014851833759417498755096, −2.77284729632779987021742093986, −1.67024146624772925732375396497, −0.47107564883467706125763451503,
0.47107564883467706125763451503, 1.67024146624772925732375396497, 2.77284729632779987021742093986, 3.99853014851833759417498755096, 5.05525167415767130788780065649, 5.92630099870698375050312139866, 7.01541488468522217454750251038, 7.44910989861900661481912566546, 7.948322873934812182251678547628, 8.855690596695143811938851766793
