| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s − 3·11-s − 12-s + 4·13-s + 16-s + 3·17-s + 2·18-s − 5·19-s + 3·22-s + 6·23-s + 24-s − 4·26-s + 5·27-s − 2·31-s − 32-s + 3·33-s − 3·34-s − 2·36-s + 2·37-s + 5·38-s − 4·39-s + 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 0.784·26-s + 0.962·27-s − 0.359·31-s − 0.176·32-s + 0.522·33-s − 0.514·34-s − 1/3·36-s + 0.328·37-s + 0.811·38-s − 0.640·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.521585518702256458221706662325, −7.993773173422627541887367381542, −7.02301401808588593198377416434, −6.22236976774475581546663735816, −5.61328186020314811053228254285, −4.75932804332696589670523572045, −3.45948532224630523972346946948, −2.60475259048520463737512973942, −1.28122616800987786392826491244, 0,
1.28122616800987786392826491244, 2.60475259048520463737512973942, 3.45948532224630523972346946948, 4.75932804332696589670523572045, 5.61328186020314811053228254285, 6.22236976774475581546663735816, 7.02301401808588593198377416434, 7.993773173422627541887367381542, 8.521585518702256458221706662325
