| L(s) = 1 | + 1.64·2-s + 2.75·3-s + 0.718·4-s + 4.35·5-s + 4.53·6-s − 3.72·7-s − 2.11·8-s + 4.57·9-s + 7.18·10-s + 2.79·11-s + 1.97·12-s − 4.88·13-s − 6.14·14-s + 11.9·15-s − 4.92·16-s + 17-s + 7.53·18-s − 2.51·19-s + 3.12·20-s − 10.2·21-s + 4.61·22-s + 4.96·23-s − 5.81·24-s + 13.9·25-s − 8.05·26-s + 4.32·27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 1.58·3-s + 0.359·4-s + 1.94·5-s + 1.85·6-s − 1.40·7-s − 0.747·8-s + 1.52·9-s + 2.27·10-s + 0.843·11-s + 0.570·12-s − 1.35·13-s − 1.64·14-s + 3.09·15-s − 1.23·16-s + 0.242·17-s + 1.77·18-s − 0.577·19-s + 0.699·20-s − 2.23·21-s + 0.983·22-s + 1.03·23-s − 1.18·24-s + 2.79·25-s − 1.57·26-s + 0.831·27-s − 0.505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.080757505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.080757505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 3 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 - 4.35T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 - 0.482T + 29T^{2} \) |
| 31 | \( 1 + 0.697T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 61 | \( 1 - 0.565T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 1.06T + 83T^{2} \) |
| 89 | \( 1 + 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.32T + 97T^{2} \) |
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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
−9.753579692866041022848078325233, −9.176615246998071282359081828862, −8.779347523675679564398891145671, −7.03946053701611243419908977322, −6.54393071931138910704589464333, −5.60088760519570637586603275641, −4.63592943788392962634763686904, −3.37029277271948437095357395024, −2.84034576294586412544258511325, −1.95346433778844280456983943666,
1.95346433778844280456983943666, 2.84034576294586412544258511325, 3.37029277271948437095357395024, 4.63592943788392962634763686904, 5.60088760519570637586603275641, 6.54393071931138910704589464333, 7.03946053701611243419908977322, 8.779347523675679564398891145671, 9.176615246998071282359081828862, 9.753579692866041022848078325233
