L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 0.236·7-s − 8-s + 9-s − 4.23·11-s − 12-s + 3·13-s + 0.236·14-s + 16-s + 7.70·17-s − 18-s + 6.47·19-s + 0.236·21-s + 4.23·22-s + 7.23·23-s + 24-s − 3·26-s − 27-s − 0.236·28-s − 6·29-s + 31-s − 32-s + 4.23·33-s − 7.70·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.0892·7-s − 0.353·8-s + 0.333·9-s − 1.27·11-s − 0.288·12-s + 0.832·13-s + 0.0630·14-s + 0.250·16-s + 1.86·17-s − 0.235·18-s + 1.48·19-s + 0.0515·21-s + 0.903·22-s + 1.50·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.0446·28-s − 1.11·29-s + 0.179·31-s − 0.176·32-s + 0.737·33-s − 1.32·34-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
\(L(1)\) |
\(\approx\) |
\(1.140881768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.140881768\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 - 8.70T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.155386492877393226445274666905, −7.61076673393195911798205228807, −7.10563981522075593598182184050, −6.04297768458086881353029597336, −5.47609632087294876885043366695, −4.90620423853925646989330986255, −3.44756085426040475445427443571, −2.99266445025011647175432175301, −1.56723637227083428126394096846, −0.72314231784000563974949183946,
0.72314231784000563974949183946, 1.56723637227083428126394096846, 2.99266445025011647175432175301, 3.44756085426040475445427443571, 4.90620423853925646989330986255, 5.47609632087294876885043366695, 6.04297768458086881353029597336, 7.10563981522075593598182184050, 7.61076673393195911798205228807, 8.155386492877393226445274666905