| L(s) = 1 | − 5-s + 11-s + 6·13-s + 2·17-s + 6·23-s + 25-s − 6·29-s − 2·31-s + 10·37-s − 8·41-s + 8·43-s − 4·47-s − 6·53-s − 55-s + 6·59-s + 8·61-s − 6·65-s − 14·67-s − 8·71-s + 2·73-s + 16·79-s − 12·83-s − 2·85-s − 18·89-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 1.66·13-s + 0.485·17-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 1.64·37-s − 1.24·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s − 0.134·55-s + 0.781·59-s + 1.02·61-s − 0.744·65-s − 1.71·67-s − 0.949·71-s + 0.234·73-s + 1.80·79-s − 1.31·83-s − 0.216·85-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.84969401625678, −12.16610309464549, −11.72440506406212, −11.23031278573080, −10.97845162878778, −10.65950711669270, −9.887701807508613, −9.478374066607666, −9.082646377898084, −8.469813386958480, −8.301340052677213, −7.620055283983426, −7.230512432315936, −6.714123214366341, −6.219676891789262, −5.696378541312088, −5.374501758329274, −4.599620672784523, −4.172875038858059, −3.670141759722050, −3.222536856576692, −2.748494231750632, −1.892637580038952, −1.281796617217236, −0.8944562147916746, 0,
0.8944562147916746, 1.281796617217236, 1.892637580038952, 2.748494231750632, 3.222536856576692, 3.670141759722050, 4.172875038858059, 4.599620672784523, 5.374501758329274, 5.696378541312088, 6.219676891789262, 6.714123214366341, 7.230512432315936, 7.620055283983426, 8.301340052677213, 8.469813386958480, 9.082646377898084, 9.478374066607666, 9.887701807508613, 10.65950711669270, 10.97845162878778, 11.23031278573080, 11.72440506406212, 12.16610309464549, 12.84969401625678
