L(s) = 1 | + 1.30·2-s + 3.02·3-s − 0.296·4-s + 0.572·5-s + 3.94·6-s + 1.21·7-s − 2.99·8-s + 6.12·9-s + 0.747·10-s − 3.82·11-s − 0.896·12-s + 1.55·13-s + 1.58·14-s + 1.73·15-s − 3.31·16-s + 3.24·17-s + 7.99·18-s + 7.65·19-s − 0.169·20-s + 3.67·21-s − 4.98·22-s − 23-s − 9.05·24-s − 4.67·25-s + 2.03·26-s + 9.44·27-s − 0.361·28-s + ⋯ |
L(s) = 1 | + 0.922·2-s + 1.74·3-s − 0.148·4-s + 0.256·5-s + 1.60·6-s + 0.460·7-s − 1.05·8-s + 2.04·9-s + 0.236·10-s − 1.15·11-s − 0.258·12-s + 0.431·13-s + 0.424·14-s + 0.446·15-s − 0.829·16-s + 0.786·17-s + 1.88·18-s + 1.75·19-s − 0.0380·20-s + 0.802·21-s − 1.06·22-s − 0.208·23-s − 1.84·24-s − 0.934·25-s + 0.398·26-s + 1.81·27-s − 0.0682·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.762012334\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.762012334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 - 0.572T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 0.392T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 0.852T + 79T^{2} \) |
| 83 | \( 1 + 0.606T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−10.20342286238482650140948248948, −9.578080658537211093353620470023, −8.709213857086559955337786948346, −7.979009290062013358483508911237, −7.24586776270401558336885951573, −5.65261280991011002502375528546, −4.93886620378656797251864932653, −3.61698787746960577334293402357, −3.15603630959753957330193980861, −1.86981339596634775435412588347,
1.86981339596634775435412588347, 3.15603630959753957330193980861, 3.61698787746960577334293402357, 4.93886620378656797251864932653, 5.65261280991011002502375528546, 7.24586776270401558336885951573, 7.979009290062013358483508911237, 8.709213857086559955337786948346, 9.578080658537211093353620470023, 10.20342286238482650140948248948