L(s) = 1 | − 3-s − 2·4-s + 5-s − 7-s + 9-s + 4·11-s + 2·12-s + 13-s − 15-s + 4·16-s + 3·17-s − 8·19-s − 2·20-s + 21-s + 25-s − 27-s + 2·28-s + 5·29-s − 2·31-s − 4·33-s − 35-s − 2·36-s + 6·37-s − 39-s + 4·41-s + 2·43-s − 8·44-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s + 0.727·17-s − 1.83·19-s − 0.447·20-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.928·29-s − 0.359·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s + 0.986·37-s − 0.160·39-s + 0.624·41-s + 0.304·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.60380554563208, −14.15466749919617, −13.59657828572664, −13.11697905637357, −12.58969799233570, −12.26193783529703, −11.72236829994126, −10.93982342987630, −10.50252774439719, −10.06031588012442, −9.456925857793489, −8.945135460319388, −8.696573940195306, −7.871022785551941, −7.312848349821038, −6.476975613722434, −6.124214158834295, −5.771574022330735, −4.943165762117033, −4.274548217711278, −4.080611738363154, −3.253430056452243, −2.447748374295660, −1.462221060524436, −0.9271383964786855, 0,
0.9271383964786855, 1.462221060524436, 2.447748374295660, 3.253430056452243, 4.080611738363154, 4.274548217711278, 4.943165762117033, 5.771574022330735, 6.124214158834295, 6.476975613722434, 7.312848349821038, 7.871022785551941, 8.696573940195306, 8.945135460319388, 9.456925857793489, 10.06031588012442, 10.50252774439719, 10.93982342987630, 11.72236829994126, 12.26193783529703, 12.58969799233570, 13.11697905637357, 13.59657828572664, 14.15466749919617, 14.60380554563208