L(s) = 1 | + 2-s + 0.321·3-s + 4-s − 5-s + 0.321·6-s − 2.34·7-s + 8-s − 2.89·9-s − 10-s + 4.30·11-s + 0.321·12-s − 0.151·13-s − 2.34·14-s − 0.321·15-s + 16-s − 2.90·17-s − 2.89·18-s + 0.850·19-s − 20-s − 0.752·21-s + 4.30·22-s + 1.17·23-s + 0.321·24-s + 25-s − 0.151·26-s − 1.89·27-s − 2.34·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.185·3-s + 0.5·4-s − 0.447·5-s + 0.131·6-s − 0.886·7-s + 0.353·8-s − 0.965·9-s − 0.316·10-s + 1.29·11-s + 0.0926·12-s − 0.0420·13-s − 0.626·14-s − 0.0828·15-s + 0.250·16-s − 0.703·17-s − 0.682·18-s + 0.195·19-s − 0.223·20-s − 0.164·21-s + 0.918·22-s + 0.245·23-s + 0.0655·24-s + 0.200·25-s − 0.0297·26-s − 0.364·27-s − 0.443·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.518380536\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.518380536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 0.321T + 3T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 0.151T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 0.850T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 - 2.16T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 8.77T + 59T^{2} \) |
| 61 | \( 1 - 4.12T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−8.066888095704430057785820744986, −7.12969568695681781909237667565, −6.62269817440468451691782633483, −5.97157416499861214676045421512, −5.25837521960849593105147976108, −4.19402028236739235043078512533, −3.73058959517439037932430699956, −2.96065267164524590167060287185, −2.14609695380517349021315788618, −0.72353793415966972735861705686,
0.72353793415966972735861705686, 2.14609695380517349021315788618, 2.96065267164524590167060287185, 3.73058959517439037932430699956, 4.19402028236739235043078512533, 5.25837521960849593105147976108, 5.97157416499861214676045421512, 6.62269817440468451691782633483, 7.12969568695681781909237667565, 8.066888095704430057785820744986