| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s + 11-s + 2·15-s − 3·17-s − 6·19-s − 4·21-s + 4·23-s − 4·25-s − 4·27-s + 29-s + 4·31-s + 2·33-s − 2·35-s + 9·37-s − 41-s + 4·43-s + 45-s − 6·47-s − 3·49-s − 6·51-s − 9·53-s + 55-s − 12·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s − 0.727·17-s − 1.37·19-s − 0.872·21-s + 0.834·23-s − 4/5·25-s − 0.769·27-s + 0.185·29-s + 0.718·31-s + 0.348·33-s − 0.338·35-s + 1.47·37-s − 0.156·41-s + 0.609·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.840·51-s − 1.23·53-s + 0.134·55-s − 1.58·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.778223673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.778223673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.93296230537174, −14.68410658425128, −14.13390869283171, −13.43541259747108, −13.20187112188589, −12.77138558591682, −12.05459654960267, −11.28027918920845, −10.90057817032416, −10.06212662518951, −9.636902357256388, −9.179776860912837, −8.689459615298315, −8.128654638879505, −7.597660827810409, −6.761811081704736, −6.292239500017994, −5.882912848695635, −4.773624127028258, −4.320040634780332, −3.497225863608780, −2.997569354636752, −2.286661606511304, −1.801468067248618, −0.5662852655779478,
0.5662852655779478, 1.801468067248618, 2.286661606511304, 2.997569354636752, 3.497225863608780, 4.320040634780332, 4.773624127028258, 5.882912848695635, 6.292239500017994, 6.761811081704736, 7.597660827810409, 8.128654638879505, 8.689459615298315, 9.179776860912837, 9.636902357256388, 10.06212662518951, 10.90057817032416, 11.28027918920845, 12.05459654960267, 12.77138558591682, 13.20187112188589, 13.43541259747108, 14.13390869283171, 14.68410658425128, 14.93296230537174
