L(s) = 1 | − 3·3-s + 6·9-s + 5·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 9·27-s + 9·29-s − 3·31-s − 8·37-s − 15·39-s − 3·41-s − 8·43-s − 7·47-s − 18·51-s − 2·53-s + 18·57-s − 4·59-s + 10·61-s + 8·67-s − 3·69-s + 7·71-s − 9·73-s + 15·75-s − 6·79-s + 9·81-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 2·9-s + 1.38·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.73·27-s + 1.67·29-s − 0.538·31-s − 1.31·37-s − 2.40·39-s − 0.468·41-s − 1.21·43-s − 1.02·47-s − 2.52·51-s − 0.274·53-s + 2.38·57-s − 0.520·59-s + 1.28·61-s + 0.977·67-s − 0.361·69-s + 0.830·71-s − 1.05·73-s + 1.73·75-s − 0.675·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.04197497337152687820054834122, −6.58816593954710107070945478976, −5.99955004194896901651448781012, −5.43923956112430414616921048860, −4.80224422790616237388517868450, −3.97526882511140771304727543873, −3.29167515147874786207239858147, −1.80587464224290004012260590916, −1.08088799825528163961182664494, 0,
1.08088799825528163961182664494, 1.80587464224290004012260590916, 3.29167515147874786207239858147, 3.97526882511140771304727543873, 4.80224422790616237388517868450, 5.43923956112430414616921048860, 5.99955004194896901651448781012, 6.58816593954710107070945478976, 7.04197497337152687820054834122