| L(s) = 1 | − 3·3-s − 6.30·5-s − 7·7-s + 9·9-s − 48.9·11-s − 2.60·13-s + 18.9·15-s + 136.·17-s − 45.2·19-s + 21·21-s + 38.1·23-s − 85.2·25-s − 27·27-s + 52.7·29-s + 14.7·31-s + 146.·33-s + 44.1·35-s + 333.·37-s + 7.82·39-s + 227.·41-s + 398.·43-s − 56.7·45-s + 184.·47-s + 49·49-s − 410.·51-s + 359.·53-s + 308.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.563·5-s − 0.377·7-s + 0.333·9-s − 1.34·11-s − 0.0556·13-s + 0.325·15-s + 1.95·17-s − 0.545·19-s + 0.218·21-s + 0.345·23-s − 0.682·25-s − 0.192·27-s + 0.337·29-s + 0.0856·31-s + 0.774·33-s + 0.213·35-s + 1.48·37-s + 0.0321·39-s + 0.865·41-s + 1.41·43-s − 0.187·45-s + 0.572·47-s + 0.142·49-s − 1.12·51-s + 0.932·53-s + 0.755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.061508509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061508509\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 6.30T + 125T^{2} \) |
| 11 | \( 1 + 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.60T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 14.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 836.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 201.T + 9.12e5T^{2} \) |
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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
−11.05476892024899824471128444071, −10.33070418562649540252627902002, −9.450090557527880109447794445201, −7.983283756026743002968805068478, −7.50339840224651441394702483979, −6.10227937039976936883793034324, −5.27478869392472973071091551291, −4.02279590241151120267156786980, −2.71230447774277872997601667317, −0.70873087346276910385381314990,
0.70873087346276910385381314990, 2.71230447774277872997601667317, 4.02279590241151120267156786980, 5.27478869392472973071091551291, 6.10227937039976936883793034324, 7.50339840224651441394702483979, 7.983283756026743002968805068478, 9.450090557527880109447794445201, 10.33070418562649540252627902002, 11.05476892024899824471128444071
