L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s − 11-s − 13-s − 3·15-s − 17-s − 6·19-s − 21-s − 2·23-s + 4·25-s + 27-s + 2·29-s − 33-s + 3·35-s − 5·37-s − 39-s + 4·41-s + 9·43-s − 3·45-s + 49-s − 51-s − 11·53-s + 3·55-s − 6·57-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.774·15-s − 0.242·17-s − 1.37·19-s − 0.218·21-s − 0.417·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s − 0.174·33-s + 0.507·35-s − 0.821·37-s − 0.160·39-s + 0.624·41-s + 1.37·43-s − 0.447·45-s + 1/7·49-s − 0.140·51-s − 1.51·53-s + 0.404·55-s − 0.794·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7377844084\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7377844084\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.47824663186652, −15.06326089889884, −14.40789422316260, −14.04852252254674, −13.17810637185583, −12.79029284046572, −12.24711470979065, −11.82994193856732, −10.93586412200157, −10.74209176479328, −9.962794227977560, −9.318921148887913, −8.681542755987569, −8.257780959689219, −7.652546070147754, −7.229753010999401, −6.505000907715964, −5.899253008329120, −4.905436442506254, −4.257799101913611, −3.925635545255822, −3.066442040879787, −2.531060877970850, −1.579596531718236, −0.3262582033069080,
0.3262582033069080, 1.579596531718236, 2.531060877970850, 3.066442040879787, 3.925635545255822, 4.257799101913611, 4.905436442506254, 5.899253008329120, 6.505000907715964, 7.229753010999401, 7.652546070147754, 8.257780959689219, 8.681542755987569, 9.318921148887913, 9.962794227977560, 10.74209176479328, 10.93586412200157, 11.82994193856732, 12.24711470979065, 12.79029284046572, 13.17810637185583, 14.04852252254674, 14.40789422316260, 15.06326089889884, 15.47824663186652