L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s + 13-s + 6·17-s + 4·21-s − 8·23-s − 27-s − 6·29-s + 4·31-s − 4·33-s − 2·37-s − 39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s − 6·51-s − 2·53-s + 12·59-s + 2·61-s − 4·63-s + 16·67-s + 8·69-s + 8·71-s + 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s + 1.56·59-s + 0.256·61-s − 0.503·63-s + 1.95·67-s + 0.963·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.39860946572274, −14.05142433357568, −13.46842193124464, −12.91157471567830, −12.45328368174750, −11.96204763264048, −11.70035414607062, −11.00810095333504, −10.30701581468492, −9.798890341364036, −9.671875725038749, −9.058629419425968, −8.184637558139810, −7.900266562564233, −6.936211710002378, −6.590001250567801, −6.279538723095010, −5.518887272907231, −5.224512294532451, −4.084738632336930, −3.712392211930468, −3.382704099655939, −2.398351502065888, −1.592741889705337, −0.8309316507954371, 0,
0.8309316507954371, 1.592741889705337, 2.398351502065888, 3.382704099655939, 3.712392211930468, 4.084738632336930, 5.224512294532451, 5.518887272907231, 6.279538723095010, 6.590001250567801, 6.936211710002378, 7.900266562564233, 8.184637558139810, 9.058629419425968, 9.671875725038749, 9.798890341364036, 10.30701581468492, 11.00810095333504, 11.70035414607062, 11.96204763264048, 12.45328368174750, 12.91157471567830, 13.46842193124464, 14.05142433357568, 14.39860946572274