| L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·13-s + 17-s − 2·21-s − 5·25-s + 27-s + 2·29-s + 8·31-s − 2·37-s − 4·39-s + 2·41-s + 8·43-s − 6·47-s − 3·49-s + 51-s + 10·53-s + 6·59-s − 6·61-s − 2·63-s − 16·67-s − 2·73-s − 5·75-s − 14·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.242·17-s − 0.436·21-s − 25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 0.640·39-s + 0.312·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 1.37·53-s + 0.781·59-s − 0.768·61-s − 0.251·63-s − 1.95·67-s − 0.234·73-s − 0.577·75-s − 1.57·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.96823167685265, −13.47209596676501, −13.20752650532795, −12.42389370791006, −12.21839710105010, −11.69482097382959, −11.08479223696806, −10.28953412100471, −10.01552765268662, −9.676751706656538, −9.067446578784063, −8.566218458606937, −8.021619263845673, −7.386027309643675, −7.170750673516039, −6.368531792974816, −5.977604808509383, −5.319925842785494, −4.553946851549740, −4.246827355562044, −3.428462807057156, −2.923734394634591, −2.444057570576988, −1.744498949625784, −0.8460364685210855, 0,
0.8460364685210855, 1.744498949625784, 2.444057570576988, 2.923734394634591, 3.428462807057156, 4.246827355562044, 4.553946851549740, 5.319925842785494, 5.977604808509383, 6.368531792974816, 7.170750673516039, 7.386027309643675, 8.021619263845673, 8.566218458606937, 9.067446578784063, 9.676751706656538, 10.01552765268662, 10.28953412100471, 11.08479223696806, 11.69482097382959, 12.21839710105010, 12.42389370791006, 13.20752650532795, 13.47209596676501, 13.96823167685265
