| L(s) = 1 | − 1.30·2-s + 3-s − 0.296·4-s + 5-s − 1.30·6-s + 1.84·7-s + 2.99·8-s + 9-s − 1.30·10-s + 1.70·11-s − 0.296·12-s − 4.90·13-s − 2.41·14-s + 15-s − 3.31·16-s + 5.70·17-s − 1.30·18-s − 7.54·19-s − 0.296·20-s + 1.84·21-s − 2.22·22-s + 3.07·23-s + 2.99·24-s + 25-s + 6.40·26-s + 27-s − 0.547·28-s + ⋯ |
| L(s) = 1 | − 0.922·2-s + 0.577·3-s − 0.148·4-s + 0.447·5-s − 0.532·6-s + 0.698·7-s + 1.05·8-s + 0.333·9-s − 0.412·10-s + 0.513·11-s − 0.0855·12-s − 1.36·13-s − 0.644·14-s + 0.258·15-s − 0.829·16-s + 1.38·17-s − 0.307·18-s − 1.73·19-s − 0.0662·20-s + 0.403·21-s − 0.473·22-s + 0.641·23-s + 0.611·24-s + 0.200·25-s + 1.25·26-s + 0.192·27-s − 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551097618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551097618\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 7.54T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 6.61T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.034272736510171124073734798004, −7.62277316292735596805652650790, −7.06151492138700332814804709953, −5.96472393352371514574173943990, −5.13070910660360390056032002652, −4.43569766222272082949371045188, −3.68721295574255194746850138189, −2.39449706592353710319165153963, −1.81751254438474036834724112169, −0.75282023593449637176705085815,
0.75282023593449637176705085815, 1.81751254438474036834724112169, 2.39449706592353710319165153963, 3.68721295574255194746850138189, 4.43569766222272082949371045188, 5.13070910660360390056032002652, 5.96472393352371514574173943990, 7.06151492138700332814804709953, 7.62277316292735596805652650790, 8.034272736510171124073734798004
