| L(s) = 1 | + 2.66e4·3-s − 1.95e6·5-s + 3.98e7·7-s − 4.53e8·9-s + 1.01e10·11-s − 2.69e10·13-s − 5.19e10·15-s − 8.01e10·17-s + 1.16e12·19-s + 1.06e12·21-s − 1.37e13·23-s + 3.81e12·25-s − 4.30e13·27-s + 6.53e13·29-s + 8.92e12·31-s + 2.70e14·33-s − 7.78e13·35-s + 5.25e14·37-s − 7.18e14·39-s − 2.63e15·41-s + 1.50e15·43-s + 8.85e14·45-s + 3.65e15·47-s − 9.80e15·49-s − 2.13e15·51-s + 4.33e16·53-s − 1.98e16·55-s + ⋯ |
| L(s) = 1 | + 0.780·3-s − 0.447·5-s + 0.373·7-s − 0.390·9-s + 1.29·11-s − 0.705·13-s − 0.349·15-s − 0.163·17-s + 0.831·19-s + 0.291·21-s − 1.59·23-s + 1/5·25-s − 1.08·27-s + 0.836·29-s + 0.0606·31-s + 1.01·33-s − 0.167·35-s + 0.664·37-s − 0.550·39-s − 1.25·41-s + 0.455·43-s + 0.174·45-s + 0.475·47-s − 0.860·49-s − 0.128·51-s + 1.80·53-s − 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{9} T \) |
| good | 3 | \( 1 - 986 p^{3} T + p^{19} T^{2} \) |
| 7 | \( 1 - 5697718 p T + p^{19} T^{2} \) |
| 11 | \( 1 - 10161579168 T + p^{19} T^{2} \) |
| 13 | \( 1 + 26970649702 T + p^{19} T^{2} \) |
| 17 | \( 1 + 4714985478 p T + p^{19} T^{2} \) |
| 19 | \( 1 - 1169772071260 T + p^{19} T^{2} \) |
| 23 | \( 1 + 13795883851698 T + p^{19} T^{2} \) |
| 29 | \( 1 - 65324757765390 T + p^{19} T^{2} \) |
| 31 | \( 1 - 8926539984748 T + p^{19} T^{2} \) |
| 37 | \( 1 - 525454617064394 T + p^{19} T^{2} \) |
| 41 | \( 1 + 2635226882131818 T + p^{19} T^{2} \) |
| 43 | \( 1 - 1501708702325062 T + p^{19} T^{2} \) |
| 47 | \( 1 - 3651608570665986 T + p^{19} T^{2} \) |
| 53 | \( 1 - 43306800238889538 T + p^{19} T^{2} \) |
| 59 | \( 1 + 51652090463616180 T + p^{19} T^{2} \) |
| 61 | \( 1 - 45200043953043002 T + p^{19} T^{2} \) |
| 67 | \( 1 + 322077213275888894 T + p^{19} T^{2} \) |
| 71 | \( 1 + 393293311705873692 T + p^{19} T^{2} \) |
| 73 | \( 1 + 672469661893471342 T + p^{19} T^{2} \) |
| 79 | \( 1 - 482639101471927720 T + p^{19} T^{2} \) |
| 83 | \( 1 - 313265345629507302 T + p^{19} T^{2} \) |
| 89 | \( 1 + 4230101056729722390 T + p^{19} T^{2} \) |
| 97 | \( 1 - 354705113301714434 T + p^{19} T^{2} \) |
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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
−10.08076832736699974222692882191, −9.053972511904482001043370959318, −8.194028756172979561058297319544, −7.23676785932041801167103069320, −5.93764225112735767804365205173, −4.51459577624604396102824799413, −3.56037969057132223041069752397, −2.47804353262236096771582235945, −1.33731007212906744202329483633, 0,
1.33731007212906744202329483633, 2.47804353262236096771582235945, 3.56037969057132223041069752397, 4.51459577624604396102824799413, 5.93764225112735767804365205173, 7.23676785932041801167103069320, 8.194028756172979561058297319544, 9.053972511904482001043370959318, 10.08076832736699974222692882191
