L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 3·11-s + 12-s + 13-s + 14-s + 16-s + 3·17-s − 18-s + 7·19-s − 21-s − 3·22-s − 3·23-s − 24-s − 5·25-s − 26-s + 27-s − 28-s − 2·31-s − 32-s + 3·33-s − 3·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.60·19-s − 0.218·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.359·31-s − 0.176·32-s + 0.522·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.025062458\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.025062458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.84917530898505951749520062866, −7.29196206242998009790294090073, −6.62205144060467217136644315994, −5.85451286877138147944599491737, −5.17037718444691816399418627505, −3.84786965322332336706423685717, −3.57388002436191549162483476002, −2.57709246508745328982871460201, −1.65167110695740874589895560514, −0.794771483853598824812401661620,
0.794771483853598824812401661620, 1.65167110695740874589895560514, 2.57709246508745328982871460201, 3.57388002436191549162483476002, 3.84786965322332336706423685717, 5.17037718444691816399418627505, 5.85451286877138147944599491737, 6.62205144060467217136644315994, 7.29196206242998009790294090073, 7.84917530898505951749520062866