| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 11-s + 12-s − 13-s − 14-s + 16-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s − 31-s − 32-s + 33-s + 37-s − 39-s − 41-s − 42-s + 44-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 11-s + 12-s − 13-s − 14-s + 16-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s − 31-s − 32-s + 33-s + 37-s − 39-s − 41-s − 42-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9157975762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9157975762\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.37921869441011245738889142431, −9.511010005843341443851104686247, −8.868716737355728499862369022957, −8.148404660797174007245273401215, −7.49922333010491836577656038357, −6.54218389364111010314616188831, −5.26987348899802349293227188662, −3.87882397059735457716023899844, −2.63845126578247891358973399219, −1.66567777863511083269438526980,
1.66567777863511083269438526980, 2.63845126578247891358973399219, 3.87882397059735457716023899844, 5.26987348899802349293227188662, 6.54218389364111010314616188831, 7.49922333010491836577656038357, 8.148404660797174007245273401215, 8.868716737355728499862369022957, 9.511010005843341443851104686247, 10.37921869441011245738889142431
