| L(s) = 1 | + 2.41·2-s − 3-s + 3.82·4-s + 2.82·5-s − 2.41·6-s − 3·7-s + 4.41·8-s + 9-s + 6.82·10-s + 0.828·11-s − 3.82·12-s + 6.65·13-s − 7.24·14-s − 2.82·15-s + 2.99·16-s + 2.41·18-s − 3·19-s + 10.8·20-s + 3·21-s + 1.99·22-s + 2.82·23-s − 4.41·24-s + 3.00·25-s + 16.0·26-s − 27-s − 11.4·28-s + 3.17·29-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.91·4-s + 1.26·5-s − 0.985·6-s − 1.13·7-s + 1.56·8-s + 0.333·9-s + 2.15·10-s + 0.249·11-s − 1.10·12-s + 1.84·13-s − 1.93·14-s − 0.730·15-s + 0.749·16-s + 0.569·18-s − 0.688·19-s + 2.42·20-s + 0.654·21-s + 0.426·22-s + 0.589·23-s − 0.901·24-s + 0.600·25-s + 3.15·26-s − 0.192·27-s − 2.17·28-s + 0.588·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.019594317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.019594317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 4.65T + 61T^{2} \) |
| 67 | \( 1 - T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−10.44496428143563425168167704978, −9.558055551070185763914618278485, −8.525826395096417210409372816285, −6.65152614802738337915777583574, −6.50101311421859803674637471543, −5.82114901885909603199998506020, −4.97201151629468830042027779862, −3.82554409378997834087816694599, −2.99273251190961818400701002649, −1.62890827471807654290795855308,
1.62890827471807654290795855308, 2.99273251190961818400701002649, 3.82554409378997834087816694599, 4.97201151629468830042027779862, 5.82114901885909603199998506020, 6.50101311421859803674637471543, 6.65152614802738337915777583574, 8.525826395096417210409372816285, 9.558055551070185763914618278485, 10.44496428143563425168167704978
