L(s) = 1 | + 2-s − 3-s + 4-s − 0.732·5-s − 6-s − 0.267·7-s + 8-s − 2·9-s − 0.732·10-s − 12-s + 3.46·13-s − 0.267·14-s + 0.732·15-s + 16-s + 7.46·17-s − 2·18-s − 19-s − 0.732·20-s + 0.267·21-s − 23-s − 24-s − 4.46·25-s + 3.46·26-s + 5·27-s − 0.267·28-s − 1.73·29-s + 0.732·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.327·5-s − 0.408·6-s − 0.101·7-s + 0.353·8-s − 0.666·9-s − 0.231·10-s − 0.288·12-s + 0.960·13-s − 0.0716·14-s + 0.189·15-s + 0.250·16-s + 1.81·17-s − 0.471·18-s − 0.229·19-s − 0.163·20-s + 0.0584·21-s − 0.208·23-s − 0.204·24-s − 0.892·25-s + 0.679·26-s + 0.962·27-s − 0.0506·28-s − 0.321·29-s + 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.202433125\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.202433125\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 + 0.267T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 14T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 2.19T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.048442498532947110226646357535, −7.66229978519197040150384382025, −6.60814751268699454778574233639, −5.93307109981927204350756998896, −5.53748924781498567311298036389, −4.68297727645734456090207918537, −3.64260003218086507290713055696, −3.26374825631749609900646047872, −1.99204649069875756022141574344, −0.76657129701112806050763467212,
0.76657129701112806050763467212, 1.99204649069875756022141574344, 3.26374825631749609900646047872, 3.64260003218086507290713055696, 4.68297727645734456090207918537, 5.53748924781498567311298036389, 5.93307109981927204350756998896, 6.60814751268699454778574233639, 7.66229978519197040150384382025, 8.048442498532947110226646357535