L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s − 2·9-s + 10-s − 6·11-s + 12-s + 4·13-s + 15-s + 16-s − 7·17-s − 2·18-s + 20-s − 6·22-s − 8·23-s + 24-s − 4·25-s + 4·26-s − 5·27-s + 29-s + 30-s − 5·31-s + 32-s − 6·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 1.80·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s + 0.223·20-s − 1.27·22-s − 1.66·23-s + 0.204·24-s − 4/5·25-s + 0.784·26-s − 0.962·27-s + 0.185·29-s + 0.182·30-s − 0.898·31-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.272311491914802200290959788425, −7.36548979799592327145223980319, −6.41066442040564781228541887494, −5.75572133881281499040856407735, −5.21143117667798970253574536144, −4.14215408366263436312167772689, −3.43956514126108981547791737635, −2.37697313770303178755069733648, −2.04663530612512389374681678841, 0,
2.04663530612512389374681678841, 2.37697313770303178755069733648, 3.43956514126108981547791737635, 4.14215408366263436312167772689, 5.21143117667798970253574536144, 5.75572133881281499040856407735, 6.41066442040564781228541887494, 7.36548979799592327145223980319, 8.272311491914802200290959788425