| L(s) = 1 | − 3-s + 5-s + 9-s + 2·11-s + 13-s − 15-s − 17-s + 4·19-s + 8·23-s − 4·25-s − 27-s − 3·29-s + 3·31-s − 2·33-s − 4·37-s − 39-s − 41-s + 4·43-s + 45-s + 9·47-s + 51-s + 2·55-s − 4·57-s + 11·59-s + 65-s + 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.538·31-s − 0.348·33-s − 0.657·37-s − 0.160·39-s − 0.156·41-s + 0.609·43-s + 0.149·45-s + 1.31·47-s + 0.140·51-s + 0.269·55-s − 0.529·57-s + 1.43·59-s + 0.124·65-s + 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.134267371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.134267371\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.38886567499482, −12.72594777452722, −12.37987145139017, −11.74781339203622, −11.35417461189111, −11.07284283095882, −10.33394485782331, −10.06145619160588, −9.335499449655883, −9.136325666325353, −8.561268650858639, −7.901900380549668, −7.243591996178201, −6.988868138415764, −6.367236639420176, −5.882594514888415, −5.349426263467415, −4.997914228621112, −4.270948443228759, −3.723881937318699, −3.196645353411565, −2.406953858611816, −1.832804181346766, −1.046543892950999, −0.6339690592003659,
0.6339690592003659, 1.046543892950999, 1.832804181346766, 2.406953858611816, 3.196645353411565, 3.723881937318699, 4.270948443228759, 4.997914228621112, 5.349426263467415, 5.882594514888415, 6.367236639420176, 6.988868138415764, 7.243591996178201, 7.901900380549668, 8.561268650858639, 9.136325666325353, 9.335499449655883, 10.06145619160588, 10.33394485782331, 11.07284283095882, 11.35417461189111, 11.74781339203622, 12.37987145139017, 12.72594777452722, 13.38886567499482
