L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 5·11-s + 12-s − 5·13-s + 15-s + 16-s + 17-s + 18-s − 4·19-s + 20-s − 5·22-s − 6·23-s + 24-s − 4·25-s − 5·26-s + 27-s + 30-s − 2·31-s + 32-s − 5·33-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 + 1/3·9-s + 0.316·10-s − 1.50·11-s + 0.288·12-s − 1.38·13-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.06·22-s − 1.25·23-s + 0.204·24-s − 4/5·25-s − 0.980·26-s + 0.192·27-s + 0.182·30-s − 0.359·31-s + 0.176·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.79483431997803457705923678715, −7.29047065139341172108048125003, −6.30437607017429459500426902947, −5.62101525083869622667479369366, −4.90333640502782013192920018596, −4.22522796489599679634255361284, −3.21506759792790996288258763655, −2.38997858467853414853731721447, −1.93823884812462526579521180187, 0,
1.93823884812462526579521180187, 2.38997858467853414853731721447, 3.21506759792790996288258763655, 4.22522796489599679634255361284, 4.90333640502782013192920018596, 5.62101525083869622667479369366, 6.30437607017429459500426902947, 7.29047065139341172108048125003, 7.79483431997803457705923678715