| L(s) = 1 | − 6.08e3·3-s − 1.25e6·5-s − 2.24e7·7-s − 9.21e7·9-s − 1.72e8·11-s + 2.18e9·13-s + 7.63e9·15-s + 3.01e10·17-s + 7.62e10·19-s + 1.36e11·21-s + 1.30e11·23-s + 8.12e11·25-s + 1.34e12·27-s − 8.03e11·29-s + 2.04e12·31-s + 1.04e12·33-s + 2.81e13·35-s − 3.38e13·37-s − 1.32e13·39-s + 5.32e13·41-s − 2.65e13·43-s + 1.15e14·45-s − 2.32e14·47-s + 2.72e14·49-s − 1.83e14·51-s + 1.63e14·53-s + 2.16e14·55-s + ⋯ |
| L(s) = 1 | − 0.535·3-s − 1.43·5-s − 1.47·7-s − 0.713·9-s − 0.242·11-s + 0.741·13-s + 0.769·15-s + 1.04·17-s + 1.03·19-s + 0.788·21-s + 0.347·23-s + 1.06·25-s + 0.917·27-s − 0.298·29-s + 0.430·31-s + 0.129·33-s + 2.11·35-s − 1.58·37-s − 0.396·39-s + 1.04·41-s − 0.346·43-s + 1.02·45-s − 1.42·47-s + 1.16·49-s − 0.561·51-s + 0.360·53-s + 0.348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 676 p^{2} T + p^{17} T^{2} \) |
| 5 | \( 1 + 251022 p T + p^{17} T^{2} \) |
| 7 | \( 1 + 458488 p^{2} T + p^{17} T^{2} \) |
| 11 | \( 1 + 172399692 T + p^{17} T^{2} \) |
| 13 | \( 1 - 167703802 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 30163933458 T + p^{17} T^{2} \) |
| 19 | \( 1 - 76275766060 T + p^{17} T^{2} \) |
| 23 | \( 1 - 130466597784 T + p^{17} T^{2} \) |
| 29 | \( 1 + 27694291830 p T + p^{17} T^{2} \) |
| 31 | \( 1 - 2045336056352 T + p^{17} T^{2} \) |
| 37 | \( 1 + 33855367078118 T + p^{17} T^{2} \) |
| 41 | \( 1 - 53206442755242 T + p^{17} T^{2} \) |
| 43 | \( 1 + 26590357792364 T + p^{17} T^{2} \) |
| 47 | \( 1 + 232565394320592 T + p^{17} T^{2} \) |
| 53 | \( 1 - 163277861935626 T + p^{17} T^{2} \) |
| 59 | \( 1 + 697820734313340 T + p^{17} T^{2} \) |
| 61 | \( 1 - 898968337037698 T + p^{17} T^{2} \) |
| 67 | \( 1 - 2667002109080572 T + p^{17} T^{2} \) |
| 71 | \( 1 - 3910637666678472 T + p^{17} T^{2} \) |
| 73 | \( 1 - 5855931724867274 T + p^{17} T^{2} \) |
| 79 | \( 1 + 23821740190145200 T + p^{17} T^{2} \) |
| 83 | \( 1 - 13915745478008556 T + p^{17} T^{2} \) |
| 89 | \( 1 + 30722744829110310 T + p^{17} T^{2} \) |
| 97 | \( 1 - 57649100896826978 T + p^{17} T^{2} \) |
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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
−11.17719857233955804455033055197, −9.950660333906271348685231069516, −8.633849845078401732807200749508, −7.50542083273591230259203639435, −6.37627585020833048309298576658, −5.24636499920269871193758246865, −3.62915368091630830362196191509, −3.09297008007157319008738266524, −0.837242045314734812540111847657, 0,
0.837242045314734812540111847657, 3.09297008007157319008738266524, 3.62915368091630830362196191509, 5.24636499920269871193758246865, 6.37627585020833048309298576658, 7.50542083273591230259203639435, 8.633849845078401732807200749508, 9.950660333906271348685231069516, 11.17719857233955804455033055197
