| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 2·11-s − 2·17-s − 4·19-s − 6·21-s − 23-s − 9·27-s + 29-s + 2·31-s − 6·33-s − 12·41-s + 5·43-s + 4·47-s − 3·49-s + 6·51-s + 9·53-s + 12·57-s − 8·59-s + 7·61-s + 12·63-s + 14·67-s + 3·69-s − 6·73-s + 4·77-s − 15·79-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 0.603·11-s − 0.485·17-s − 0.917·19-s − 1.30·21-s − 0.208·23-s − 1.73·27-s + 0.185·29-s + 0.359·31-s − 1.04·33-s − 1.87·41-s + 0.762·43-s + 0.583·47-s − 3/7·49-s + 0.840·51-s + 1.23·53-s + 1.58·57-s − 1.04·59-s + 0.896·61-s + 1.51·63-s + 1.71·67-s + 0.361·69-s − 0.702·73-s + 0.455·77-s − 1.68·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−15.41622404048973, −14.72364250273043, −14.33601849472652, −13.46336241919428, −13.13346671225263, −12.36309442996156, −11.96085791042286, −11.64541891284920, −10.91650607737101, −10.78268286057807, −10.06455420092866, −9.577769746145844, −8.650465473697852, −8.340539120090203, −7.413957614520016, −6.880312225847800, −6.439633872690077, −5.847809482071831, −5.306372809951551, −4.641463095301947, −4.308626289911010, −3.551094654498462, −2.339212544316706, −1.632436606206042, −0.8950861142767320, 0,
0.8950861142767320, 1.632436606206042, 2.339212544316706, 3.551094654498462, 4.308626289911010, 4.641463095301947, 5.306372809951551, 5.847809482071831, 6.439633872690077, 6.880312225847800, 7.413957614520016, 8.340539120090203, 8.650465473697852, 9.577769746145844, 10.06455420092866, 10.78268286057807, 10.91650607737101, 11.64541891284920, 11.96085791042286, 12.36309442996156, 13.13346671225263, 13.46336241919428, 14.33601849472652, 14.72364250273043, 15.41622404048973
