| L(s) = 1 | + 1.88·2-s + 3-s + 1.56·4-s + 2.17·5-s + 1.88·6-s − 3.72·7-s − 0.824·8-s + 9-s + 4.10·10-s − 11-s + 1.56·12-s − 7.03·14-s + 2.17·15-s − 4.68·16-s + 2.48·17-s + 1.88·18-s + 4.08·19-s + 3.40·20-s − 3.72·21-s − 1.88·22-s + 6.28·23-s − 0.824·24-s − 0.267·25-s + 27-s − 5.82·28-s + 9.95·29-s + 4.10·30-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.781·4-s + 0.972·5-s + 0.770·6-s − 1.40·7-s − 0.291·8-s + 0.333·9-s + 1.29·10-s − 0.301·11-s + 0.451·12-s − 1.87·14-s + 0.561·15-s − 1.17·16-s + 0.603·17-s + 0.444·18-s + 0.936·19-s + 0.760·20-s − 0.812·21-s − 0.402·22-s + 1.30·23-s − 0.168·24-s − 0.0535·25-s + 0.192·27-s − 1.10·28-s + 1.84·29-s + 0.749·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.181719836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.181719836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 - 9.95T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 8.75T + 43T^{2} \) |
| 47 | \( 1 + 0.202T + 47T^{2} \) |
| 53 | \( 1 - 3.06T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 + 2.93T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 8.11T + 97T^{2} \) |
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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
−8.092258964510449021926546605317, −7.07035445068526181452289505528, −6.56145728317134793986644282736, −5.84207206997011344324718698216, −5.32626099782444992886687572570, −4.45999859217391579034327475606, −3.57497383084160632144617478953, −2.86182483549428251962669141745, −2.53913611079974066955464306483, −0.982489664036270518341187582695,
0.982489664036270518341187582695, 2.53913611079974066955464306483, 2.86182483549428251962669141745, 3.57497383084160632144617478953, 4.45999859217391579034327475606, 5.32626099782444992886687572570, 5.84207206997011344324718698216, 6.56145728317134793986644282736, 7.07035445068526181452289505528, 8.092258964510449021926546605317
