L(s) = 1 | − 3·13-s − 2·17-s + 19-s − 2·23-s + 8·29-s − 8·31-s − 7·37-s − 8·43-s − 10·47-s − 14·53-s − 10·59-s − 7·61-s − 5·67-s − 12·71-s − 11·73-s + 7·79-s + 14·83-s − 6·89-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.832·13-s − 0.485·17-s + 0.229·19-s − 0.417·23-s + 1.48·29-s − 1.43·31-s − 1.15·37-s − 1.21·43-s − 1.45·47-s − 1.92·53-s − 1.30·59-s − 0.896·61-s − 0.610·67-s − 1.42·71-s − 1.28·73-s + 0.787·79-s + 1.53·83-s − 0.635·89-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.50135312644883, −13.35528577815603, −12.57102278165472, −12.25768913966239, −11.85796515926237, −11.33848585293390, −10.70541250270764, −10.43298630608249, −9.842338154469545, −9.376053865965213, −8.948104775324959, −8.380887785427273, −7.849732988933581, −7.455836922276592, −6.825948903952725, −6.418425588371268, −5.930270167774508, −5.104634651239521, −4.876568260110836, −4.352043746617746, −3.544531346484962, −3.116737921413932, −2.538481998369891, −1.706020759923438, −1.410189000494159, 0, 0,
1.410189000494159, 1.706020759923438, 2.538481998369891, 3.116737921413932, 3.544531346484962, 4.352043746617746, 4.876568260110836, 5.104634651239521, 5.930270167774508, 6.418425588371268, 6.825948903952725, 7.455836922276592, 7.849732988933581, 8.380887785427273, 8.948104775324959, 9.376053865965213, 9.842338154469545, 10.43298630608249, 10.70541250270764, 11.33848585293390, 11.85796515926237, 12.25768913966239, 12.57102278165472, 13.35528577815603, 13.50135312644883