| L(s) = 1 | − 3-s + 5-s + 1.87·7-s + 9-s + 3.74·11-s + 4.08·13-s − 15-s + 1.40·17-s + 3.29·19-s − 1.87·21-s + 8.85·23-s + 25-s − 27-s − 6.06·29-s − 3.60·31-s − 3.74·33-s + 1.87·35-s + 5.24·37-s − 4.08·39-s + 6.49·41-s + 9.07·43-s + 45-s + 7.62·47-s − 3.46·49-s − 1.40·51-s + 8.26·53-s + 3.74·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.710·7-s + 0.333·9-s + 1.13·11-s + 1.13·13-s − 0.258·15-s + 0.341·17-s + 0.756·19-s − 0.410·21-s + 1.84·23-s + 0.200·25-s − 0.192·27-s − 1.12·29-s − 0.646·31-s − 0.652·33-s + 0.317·35-s + 0.862·37-s − 0.654·39-s + 1.01·41-s + 1.38·43-s + 0.149·45-s + 1.11·47-s − 0.495·49-s − 0.197·51-s + 1.13·53-s + 0.505·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.796514240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.796514240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 7.62T + 47T^{2} \) |
| 53 | \( 1 - 8.26T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.29T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.58302844511783734975043898251, −7.22340375859616201123223754391, −6.29719503809259804278200131909, −5.76483051870784765451345204925, −5.18728358994971031754189639574, −4.28216209501703921340958623764, −3.67145425263624926137013959314, −2.63250044346003207822145792854, −1.39968246569389582995154719181, −1.03061172498476604525246130979,
1.03061172498476604525246130979, 1.39968246569389582995154719181, 2.63250044346003207822145792854, 3.67145425263624926137013959314, 4.28216209501703921340958623764, 5.18728358994971031754189639574, 5.76483051870784765451345204925, 6.29719503809259804278200131909, 7.22340375859616201123223754391, 7.58302844511783734975043898251
