L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 4·11-s + 12-s − 13-s − 4·14-s + 16-s − 2·17-s + 18-s − 8·19-s − 4·21-s − 4·22-s + 24-s − 26-s + 27-s − 4·28-s + 6·29-s − 4·31-s + 32-s − 4·33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.872·21-s − 0.852·22-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.669837022733871834740693269948, −8.086930166936528444515660462457, −6.88136623657690699633393946488, −6.59890751507414505352716522675, −5.55115129298543617853829902783, −4.60729137120671133863970405352, −3.69246907163843516031432566202, −2.85978826953841430658632684820, −2.15232787243142660514248344878, 0,
2.15232787243142660514248344878, 2.85978826953841430658632684820, 3.69246907163843516031432566202, 4.60729137120671133863970405352, 5.55115129298543617853829902783, 6.59890751507414505352716522675, 6.88136623657690699633393946488, 8.086930166936528444515660462457, 8.669837022733871834740693269948