L(s) = 1 | − 3-s + 2.82·5-s + 2.82·7-s + 9-s − 2·11-s − 2.82·15-s + 7.65·17-s − 2.82·19-s − 2.82·21-s + 4·23-s + 3.00·25-s − 27-s + 2·29-s − 1.17·31-s + 2·33-s + 8.00·35-s + 7.65·37-s − 5.17·41-s + 1.65·43-s + 2.82·45-s − 11.6·47-s + 1.00·49-s − 7.65·51-s − 2·53-s − 5.65·55-s + 2.82·57-s + 7.65·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.26·5-s + 1.06·7-s + 0.333·9-s − 0.603·11-s − 0.730·15-s + 1.85·17-s − 0.648·19-s − 0.617·21-s + 0.834·23-s + 0.600·25-s − 0.192·27-s + 0.371·29-s − 0.210·31-s + 0.348·33-s + 1.35·35-s + 1.25·37-s − 0.807·41-s + 0.252·43-s + 0.421·45-s − 1.70·47-s + 0.142·49-s − 1.07·51-s − 0.274·53-s − 0.762·55-s + 0.374·57-s + 0.996·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.741234433\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.741234433\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−7.990141079814693768892221126732, −7.01773987827997898989848438516, −6.35765025504011613620564501832, −5.51611287654403435657618377256, −5.27751837172300887949451336596, −4.57923561797008159184475209654, −3.46757840849293677255791493855, −2.47902930117395724218770063979, −1.68408325810760880658196138268, −0.901890986361598985393431056037,
0.901890986361598985393431056037, 1.68408325810760880658196138268, 2.47902930117395724218770063979, 3.46757840849293677255791493855, 4.57923561797008159184475209654, 5.27751837172300887949451336596, 5.51611287654403435657618377256, 6.35765025504011613620564501832, 7.01773987827997898989848438516, 7.990141079814693768892221126732