L(s) = 1 | + 2·7-s − 11-s + 4·13-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s + 6·29-s + 8·31-s + 10·37-s − 6·41-s − 8·43-s + 6·47-s − 3·49-s − 8·61-s + 4·67-s − 6·71-s + 2·73-s − 2·77-s + 14·79-s − 12·83-s + 6·89-s + 8·91-s + 14·97-s + 6·101-s − 4·103-s − 12·107-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s − 1.02·61-s + 0.488·67-s − 0.712·71-s + 0.234·73-s − 0.227·77-s + 1.57·79-s − 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.597·101-s − 0.394·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.553217758\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.553217758\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.023485538763249731063179858133, −7.62198255296856017877685592804, −6.46698528898118851301371368497, −5.93219336897033405553442715432, −5.20703819122366802812198505548, −4.43317830098000794362534321893, −3.60617492632818609088993638744, −2.83372192473594711021853554092, −1.69195435153571175377540877194, −0.895207527765570180254099809505,
0.895207527765570180254099809505, 1.69195435153571175377540877194, 2.83372192473594711021853554092, 3.60617492632818609088993638744, 4.43317830098000794362534321893, 5.20703819122366802812198505548, 5.93219336897033405553442715432, 6.46698528898118851301371368497, 7.62198255296856017877685592804, 8.023485538763249731063179858133