| L(s) = 1 | + 4-s + 9-s − 11-s − 13-s + 16-s + 2·17-s − 23-s − 31-s + 36-s − 41-s + 43-s − 44-s − 47-s + 49-s − 52-s − 53-s − 59-s + 64-s + 2·67-s + 2·68-s − 79-s + 81-s − 83-s − 92-s − 97-s − 99-s − 101-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 11-s − 13-s + 16-s + 2·17-s − 23-s − 31-s + 36-s − 41-s + 43-s − 44-s − 47-s + 49-s − 52-s − 53-s − 59-s + 64-s + 2·67-s + 2·68-s − 79-s + 81-s − 83-s − 92-s − 97-s − 99-s − 101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.306800193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306800193\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.09559903691656834320565371908, −9.628555223560151531874458364810, −8.031972291333033422187511050338, −7.62023322981695578499322140977, −6.91350403711780700669299381404, −5.78976507631527327029210430350, −5.07927653121296481059815007102, −3.74058753618653695707659150464, −2.70250907951448229719334897100, −1.59694872944907197385576404447,
1.59694872944907197385576404447, 2.70250907951448229719334897100, 3.74058753618653695707659150464, 5.07927653121296481059815007102, 5.78976507631527327029210430350, 6.91350403711780700669299381404, 7.62023322981695578499322140977, 8.031972291333033422187511050338, 9.628555223560151531874458364810, 10.09559903691656834320565371908
