L(s) = 1 | − 3-s + 9-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s − 27-s + 10·29-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s + 8·47-s − 2·51-s + 6·53-s − 4·57-s + 4·59-s − 10·61-s − 4·67-s + 16·71-s − 14·73-s − 8·79-s + 81-s − 4·83-s − 10·87-s − 10·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.192·27-s + 1.85·29-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 1.89·71-s − 1.63·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s − 1.07·87-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.11486946260899, −12.56210511780713, −12.26466879448908, −11.78694648032189, −11.20486098829895, −10.87376291741419, −10.35238770432469, −10.01387480578487, −9.534421991097415, −8.897821290306772, −8.414840423030314, −7.896127917206401, −7.444756977444472, −7.083889251453529, −6.296079443477884, −5.964794679056341, −5.435909214475125, −5.073294599849481, −4.347286584935887, −4.056975857104346, −3.112160573899546, −2.793606022893126, −2.205610434166515, −1.132445768982925, −0.9503263975924160, 0,
0.9503263975924160, 1.132445768982925, 2.205610434166515, 2.793606022893126, 3.112160573899546, 4.056975857104346, 4.347286584935887, 5.073294599849481, 5.435909214475125, 5.964794679056341, 6.296079443477884, 7.083889251453529, 7.444756977444472, 7.896127917206401, 8.414840423030314, 8.897821290306772, 9.534421991097415, 10.01387480578487, 10.35238770432469, 10.87376291741419, 11.20486098829895, 11.78694648032189, 12.26466879448908, 12.56210511780713, 13.11486946260899