| L(s) = 1 | + 2.13·2-s − 3-s + 2.55·4-s − 2.13·6-s − 2.16·7-s + 1.17·8-s + 9-s − 2.50·11-s − 2.55·12-s + 4.33·13-s − 4.62·14-s − 2.59·16-s + 6.77·17-s + 2.13·18-s + 6.83·19-s + 2.16·21-s − 5.35·22-s + 1.67·23-s − 1.17·24-s + 9.24·26-s − 27-s − 5.53·28-s + 4.44·29-s + 6.56·31-s − 7.88·32-s + 2.50·33-s + 14.4·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.27·4-s − 0.870·6-s − 0.819·7-s + 0.415·8-s + 0.333·9-s − 0.756·11-s − 0.736·12-s + 1.20·13-s − 1.23·14-s − 0.648·16-s + 1.64·17-s + 0.502·18-s + 1.56·19-s + 0.473·21-s − 1.14·22-s + 0.349·23-s − 0.239·24-s + 1.81·26-s − 0.192·27-s − 1.04·28-s + 0.826·29-s + 1.17·31-s − 1.39·32-s + 0.436·33-s + 2.47·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.199686180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.199686180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 1.11T + 73T^{2} \) |
| 79 | \( 1 - 7.78T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.436917006823789467464891849866, −8.276672578655908588512202537432, −7.30722131641447629401167328682, −6.53428996539718052915463900989, −5.67313352877031713198191964514, −5.41330731396015158787291623748, −4.32752492128477389280848744703, −3.36759856488971793483560059971, −2.86496542753730561422586226867, −1.04569297838411953444942667075,
1.04569297838411953444942667075, 2.86496542753730561422586226867, 3.36759856488971793483560059971, 4.32752492128477389280848744703, 5.41330731396015158787291623748, 5.67313352877031713198191964514, 6.53428996539718052915463900989, 7.30722131641447629401167328682, 8.276672578655908588512202537432, 9.436917006823789467464891849866
