L(s) = 1 | − 3-s − 7-s + 9-s + 11-s − 5·13-s + 6·17-s − 5·19-s + 21-s − 23-s − 27-s + 9·29-s + 9·31-s − 33-s + 5·39-s − 43-s − 8·47-s + 49-s − 6·51-s + 8·53-s + 5·57-s − 8·59-s + 14·61-s − 63-s + 12·67-s + 69-s − 15·71-s − 16·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 1.45·17-s − 1.14·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.67·29-s + 1.61·31-s − 0.174·33-s + 0.800·39-s − 0.152·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.09·53-s + 0.662·57-s − 1.04·59-s + 1.79·61-s − 0.125·63-s + 1.46·67-s + 0.120·69-s − 1.78·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.570767272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570767272\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−12.42687903911159, −11.98761673916381, −11.74463118715064, −11.31550561823928, −10.46990560547417, −10.12358492169325, −10.01604176799098, −9.573282195810257, −8.729739045006713, −8.430274311028640, −7.954024678960075, −7.314347186709637, −6.938781010265105, −6.508309626525453, −5.978472445337361, −5.577743204169508, −4.970414088335567, −4.414992551441206, −4.263640477177564, −3.300999989201227, −2.917820013685749, −2.375958422196101, −1.648209901490244, −0.9716284865392675, −0.3910411687041446,
0.3910411687041446, 0.9716284865392675, 1.648209901490244, 2.375958422196101, 2.917820013685749, 3.300999989201227, 4.263640477177564, 4.414992551441206, 4.970414088335567, 5.577743204169508, 5.978472445337361, 6.508309626525453, 6.938781010265105, 7.314347186709637, 7.954024678960075, 8.430274311028640, 8.729739045006713, 9.573282195810257, 10.01604176799098, 10.12358492169325, 10.46990560547417, 11.31550561823928, 11.74463118715064, 11.98761673916381, 12.42687903911159