| L(s) = 1 | + 2·5-s − 3·7-s + 13-s + 19-s + 4·23-s − 25-s − 8·29-s − 9·31-s − 6·35-s + 9·37-s + 2·41-s − 4·43-s + 2·47-s + 2·49-s − 2·53-s + 6·59-s + 11·61-s + 2·65-s + 15·67-s + 6·71-s − 9·73-s + 79-s − 8·89-s − 3·91-s + 2·95-s + 7·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.13·7-s + 0.277·13-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 1.48·29-s − 1.61·31-s − 1.01·35-s + 1.47·37-s + 0.312·41-s − 0.609·43-s + 0.291·47-s + 2/7·49-s − 0.274·53-s + 0.781·59-s + 1.40·61-s + 0.248·65-s + 1.83·67-s + 0.712·71-s − 1.05·73-s + 0.112·79-s − 0.847·89-s − 0.314·91-s + 0.205·95-s + 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05491680304926, −12.91872304527535, −12.52094289364635, −11.63421025927391, −11.29114744205481, −10.93716710220349, −10.21934484017617, −9.794587782708962, −9.557152909591531, −9.034246707829606, −8.711541212122672, −7.853010489290893, −7.502174851429844, −6.838790239539393, −6.529728819577427, −5.948269739647687, −5.475037972340854, −5.224438526913785, −4.321620262251457, −3.666511241986788, −3.450016926540343, −2.604339974998391, −2.211884219756977, −1.525418021852300, −0.7912401197147888, 0,
0.7912401197147888, 1.525418021852300, 2.211884219756977, 2.604339974998391, 3.450016926540343, 3.666511241986788, 4.321620262251457, 5.224438526913785, 5.475037972340854, 5.948269739647687, 6.529728819577427, 6.838790239539393, 7.502174851429844, 7.853010489290893, 8.711541212122672, 9.034246707829606, 9.557152909591531, 9.794587782708962, 10.21934484017617, 10.93716710220349, 11.29114744205481, 11.63421025927391, 12.52094289364635, 12.91872304527535, 13.05491680304926
