L(s) = 1 | − 830·5-s + 672·7-s + 7.34e4·11-s − 7.82e4·13-s + 1.61e5·17-s − 6.53e5·19-s + 1.06e6·23-s − 1.26e6·25-s − 3.82e6·29-s − 1.57e6·31-s − 5.57e5·35-s + 1.60e7·37-s − 2.62e7·41-s − 4.44e7·43-s − 1.43e7·47-s − 3.99e7·49-s + 2.43e7·53-s − 6.09e7·55-s − 1.19e7·59-s − 1.89e8·61-s + 6.49e7·65-s − 1.06e8·67-s − 3.02e8·71-s + 8.17e7·73-s + 4.93e7·77-s + 3.15e8·79-s − 7.52e8·83-s + ⋯ |
L(s) = 1 | − 0.593·5-s + 0.105·7-s + 1.51·11-s − 0.759·13-s + 0.469·17-s − 1.15·19-s + 0.794·23-s − 0.647·25-s − 1.00·29-s − 0.307·31-s − 0.0628·35-s + 1.40·37-s − 1.45·41-s − 1.98·43-s − 0.428·47-s − 0.988·49-s + 0.424·53-s − 0.898·55-s − 0.128·59-s − 1.75·61-s + 0.451·65-s − 0.646·67-s − 1.41·71-s + 0.337·73-s + 0.160·77-s + 0.910·79-s − 1.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 166 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 96 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 73468 T + p^{9} T^{2} \) |
| 13 | \( 1 + 78242 T + p^{9} T^{2} \) |
| 17 | \( 1 - 161726 T + p^{9} T^{2} \) |
| 19 | \( 1 + 653572 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1066696 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3824838 T + p^{9} T^{2} \) |
| 31 | \( 1 + 1579480 T + p^{9} T^{2} \) |
| 37 | \( 1 - 16015590 T + p^{9} T^{2} \) |
| 41 | \( 1 + 26268282 T + p^{9} T^{2} \) |
| 43 | \( 1 + 44495228 T + p^{9} T^{2} \) |
| 47 | \( 1 + 14324160 T + p^{9} T^{2} \) |
| 53 | \( 1 - 24386050 T + p^{9} T^{2} \) |
| 59 | \( 1 + 11942084 T + p^{9} T^{2} \) |
| 61 | \( 1 + 189740258 T + p^{9} T^{2} \) |
| 67 | \( 1 + 106709572 T + p^{9} T^{2} \) |
| 71 | \( 1 + 302754376 T + p^{9} T^{2} \) |
| 73 | \( 1 - 81769546 T + p^{9} T^{2} \) |
| 79 | \( 1 - 315315352 T + p^{9} T^{2} \) |
| 83 | \( 1 + 752833276 T + p^{9} T^{2} \) |
| 89 | \( 1 - 433284294 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1282496642 T + p^{9} T^{2} \) |
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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
−12.04685437961247123460140832083, −11.31229761757030501748781120242, −9.874101893585847178327941689600, −8.753836943100851189773169888051, −7.49461424692404120027818803256, −6.32320432735280900359594019856, −4.66186524437132029792808857672, −3.47286057320199859281825363907, −1.65121520812514296804937578064, 0,
1.65121520812514296804937578064, 3.47286057320199859281825363907, 4.66186524437132029792808857672, 6.32320432735280900359594019856, 7.49461424692404120027818803256, 8.753836943100851189773169888051, 9.874101893585847178327941689600, 11.31229761757030501748781120242, 12.04685437961247123460140832083