L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 2·7-s + 8-s + 9-s + 10-s + 12-s + 2·14-s + 15-s + 16-s + 2·17-s + 18-s + 2·19-s + 20-s + 2·21-s + 24-s + 25-s + 27-s + 2·28-s + 6·29-s + 30-s + 32-s + 2·34-s + 2·35-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.436·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 0.182·30-s + 0.176·32-s + 0.342·34-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.725745493\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.725745493\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−8.342028774847602354251421474555, −7.896986653487582702695944115207, −6.97045277711945804400512275099, −6.33369706725932761568797966118, −5.26983629712421846434428785352, −4.88602051994619432727972026398, −3.83834220942440075431979505126, −3.06386122887057512667863999826, −2.13539748501880631467646847021, −1.25112879593738218514279144779,
1.25112879593738218514279144779, 2.13539748501880631467646847021, 3.06386122887057512667863999826, 3.83834220942440075431979505126, 4.88602051994619432727972026398, 5.26983629712421846434428785352, 6.33369706725932761568797966118, 6.97045277711945804400512275099, 7.896986653487582702695944115207, 8.342028774847602354251421474555