L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s − 5·11-s + 12-s − 13-s − 14-s + 15-s + 16-s + 3·17-s + 18-s + 19-s + 20-s − 21-s − 5·22-s + 24-s − 4·25-s − 26-s + 27-s − 28-s + 9·29-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 − 1.50·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s − 1.06·22-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288834 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288834 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.254366407\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.254366407\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.83203183176100, −12.50657855369703, −11.78082879336143, −11.55456011423641, −10.85355033006831, −10.23392019446792, −10.02329924412695, −9.746397410329995, −9.061166097901718, −8.356597279317542, −8.055328350608480, −7.606107359290187, −7.157245123727046, −6.482865417537496, −6.062454625816554, −5.616825683040338, −4.989235145493636, −4.703039687520943, −4.018968827360359, −3.427821731722619, −2.878883428764165, −2.487647806155543, −2.133240977974470, −1.162071286427988, −0.5856319458304429,
0.5856319458304429, 1.162071286427988, 2.133240977974470, 2.487647806155543, 2.878883428764165, 3.427821731722619, 4.018968827360359, 4.703039687520943, 4.989235145493636, 5.616825683040338, 6.062454625816554, 6.482865417537496, 7.157245123727046, 7.606107359290187, 8.055328350608480, 8.356597279317542, 9.061166097901718, 9.746397410329995, 10.02329924412695, 10.23392019446792, 10.85355033006831, 11.55456011423641, 11.78082879336143, 12.50657855369703, 12.83203183176100