L(s) = 1 | + 2-s + 5-s − 8-s + 10-s + 11-s − 16-s + 17-s + 22-s + 29-s − 31-s + 34-s − 40-s + 49-s + 55-s + 58-s − 61-s − 62-s + 64-s − 67-s − 2·71-s − 80-s + 83-s + 85-s − 88-s + 98-s − 2·101-s + 2·109-s + ⋯ |
L(s) = 1 | + 2-s + 5-s − 8-s + 10-s + 11-s − 16-s + 17-s + 22-s + 29-s − 31-s + 34-s − 40-s + 49-s + 55-s + 58-s − 61-s − 62-s + 64-s − 67-s − 2·71-s − 80-s + 83-s + 85-s − 88-s + 98-s − 2·101-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.095309664\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.095309664\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.261350256466312134300660041231, −8.726784860072541168048129136605, −7.57920940932432187060207398292, −6.57116970777904417222522784581, −5.93830779433496274933950719192, −5.36044839722544646935399516069, −4.43159691693746299338325365987, −3.61352133826178070111536630164, −2.69276183990010526705050017922, −1.43858530967738514254427092353,
1.43858530967738514254427092353, 2.69276183990010526705050017922, 3.61352133826178070111536630164, 4.43159691693746299338325365987, 5.36044839722544646935399516069, 5.93830779433496274933950719192, 6.57116970777904417222522784581, 7.57920940932432187060207398292, 8.726784860072541168048129136605, 9.261350256466312134300660041231