L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 12-s − 3·13-s + 14-s − 15-s + 16-s − 5·17-s − 18-s − 3·19-s + 20-s + 21-s − 23-s + 24-s + 25-s + 3·26-s − 27-s − 28-s − 9·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.832·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.688·19-s + 0.223·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.24104268365622, −13.54747085486471, −13.20856011187777, −12.50231773668256, −12.31257143195565, −11.64322758727607, −11.02122479966994, −10.70921088640571, −10.29475471736734, −9.569519693658837, −9.261698079451236, −8.888663121764527, −8.067097929903737, −7.590353914675618, −7.035144092633661, −6.383700670248752, −6.292364977268444, −5.338941177182020, −5.057680296477040, −4.188447792354907, −3.692572234343692, −2.755151985978635, −2.146842791770889, −1.739762073007526, −0.6494438768137172, 0,
0.6494438768137172, 1.739762073007526, 2.146842791770889, 2.755151985978635, 3.692572234343692, 4.188447792354907, 5.057680296477040, 5.338941177182020, 6.292364977268444, 6.383700670248752, 7.035144092633661, 7.590353914675618, 8.067097929903737, 8.888663121764527, 9.261698079451236, 9.569519693658837, 10.29475471736734, 10.70921088640571, 11.02122479966994, 11.64322758727607, 12.31257143195565, 12.50231773668256, 13.20856011187777, 13.54747085486471, 14.24104268365622