| L(s) = 1 | − 0.364·3-s − 5-s + 0.635·7-s − 2.86·9-s + 1.63·11-s − 13-s + 0.364·15-s − 6.59·17-s − 0.231·21-s + 0.867·23-s + 25-s + 2.13·27-s − 9.09·29-s − 1.86·31-s − 0.596·33-s − 0.635·35-s + 0.635·37-s + 0.364·39-s + 1.90·41-s + 3.96·43-s + 2.86·45-s + 9.09·47-s − 6.59·49-s + 2.40·51-s − 9.86·53-s − 1.63·55-s + 9.09·59-s + ⋯ |
| L(s) = 1 | − 0.210·3-s − 0.447·5-s + 0.240·7-s − 0.955·9-s + 0.493·11-s − 0.277·13-s + 0.0941·15-s − 1.59·17-s − 0.0505·21-s + 0.180·23-s + 0.200·25-s + 0.411·27-s − 1.68·29-s − 0.335·31-s − 0.103·33-s − 0.107·35-s + 0.104·37-s + 0.0583·39-s + 0.297·41-s + 0.603·43-s + 0.427·45-s + 1.32·47-s − 0.942·49-s + 0.336·51-s − 1.35·53-s − 0.220·55-s + 1.18·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013686360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013686360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.364T + 3T^{2} \) |
| 7 | \( 1 - 0.635T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 23 | \( 1 - 0.867T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 - 0.635T + 37T^{2} \) |
| 41 | \( 1 - 1.90T + 41T^{2} \) |
| 43 | \( 1 - 3.96T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 0.403T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 0.542T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−7.948300031492857756088411580813, −7.17825576761900570790190928511, −6.57783503807984542893839818249, −5.79541043123975003957050152807, −5.12919842299840032953370780766, −4.30611555601756514015098336509, −3.66719398011942763209050265500, −2.66427990268088017021026272053, −1.87753436567450120209138491856, −0.49553532219374068952532041807,
0.49553532219374068952532041807, 1.87753436567450120209138491856, 2.66427990268088017021026272053, 3.66719398011942763209050265500, 4.30611555601756514015098336509, 5.12919842299840032953370780766, 5.79541043123975003957050152807, 6.57783503807984542893839818249, 7.17825576761900570790190928511, 7.948300031492857756088411580813
