L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 3·7-s − 8-s + 9-s − 11-s − 12-s − 3·13-s − 3·14-s + 16-s + 4·17-s − 18-s − 2·19-s − 3·21-s + 22-s − 4·23-s + 24-s + 3·26-s − 27-s + 3·28-s + 6·29-s − 31-s − 32-s + 33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.458·19-s − 0.654·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s + 0.566·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.977334643115892156564743811145, −7.39021320700378986203084177735, −6.60496882593567599890947304296, −5.75456496111527370488750179630, −5.04742557786122364222137520764, −4.40537670844249671978213548979, −3.19703856010765032274213455948, −2.10988997095340602391882969642, −1.30832563771107619400793162383, 0,
1.30832563771107619400793162383, 2.10988997095340602391882969642, 3.19703856010765032274213455948, 4.40537670844249671978213548979, 5.04742557786122364222137520764, 5.75456496111527370488750179630, 6.60496882593567599890947304296, 7.39021320700378986203084177735, 7.977334643115892156564743811145