| L(s) = 1 | − 1.02·3-s + 3.72·5-s + 1.42·7-s − 1.94·9-s + 0.990·11-s − 3.96·13-s − 3.82·15-s − 2.93·17-s − 0.121·19-s − 1.46·21-s − 7.15·23-s + 8.85·25-s + 5.07·27-s − 1.74·29-s − 1.73·31-s − 1.01·33-s + 5.30·35-s + 9.97·37-s + 4.06·39-s + 10.4·43-s − 7.24·45-s − 13.1·47-s − 4.96·49-s + 3.01·51-s − 8.32·53-s + 3.68·55-s + 0.124·57-s + ⋯ |
| L(s) = 1 | − 0.592·3-s + 1.66·5-s + 0.538·7-s − 0.648·9-s + 0.298·11-s − 1.09·13-s − 0.986·15-s − 0.711·17-s − 0.0278·19-s − 0.319·21-s − 1.49·23-s + 1.77·25-s + 0.977·27-s − 0.323·29-s − 0.310·31-s − 0.177·33-s + 0.897·35-s + 1.64·37-s + 0.651·39-s + 1.60·43-s − 1.08·45-s − 1.92·47-s − 0.709·49-s + 0.421·51-s − 1.14·53-s + 0.497·55-s + 0.0165·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 0.990T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 + 0.121T + 19T^{2} \) |
| 23 | \( 1 + 7.15T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 1.75T + 67T^{2} \) |
| 71 | \( 1 + 8.84T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.67300344044768993509294494400, −6.58214882383977508685994712941, −6.20492102391663316235344245631, −5.57263381515654334642618639159, −4.95014478558193923360927600877, −4.26697165428323613391185134239, −2.86490788142677842153504427812, −2.21826831041065423327816715953, −1.45253734619234442959060224750, 0,
1.45253734619234442959060224750, 2.21826831041065423327816715953, 2.86490788142677842153504427812, 4.26697165428323613391185134239, 4.95014478558193923360927600877, 5.57263381515654334642618639159, 6.20492102391663316235344245631, 6.58214882383977508685994712941, 7.67300344044768993509294494400
