| L(s) = 1 | + 6.56e3·3-s − 1.63e5·5-s + 2.08e7·7-s + 4.30e7·9-s − 8.17e8·11-s + 2.99e8·13-s − 1.07e9·15-s − 4.47e10·17-s − 7.87e10·19-s + 1.36e11·21-s + 7.04e11·23-s − 7.36e11·25-s + 2.82e11·27-s − 1.63e11·29-s − 1.04e12·31-s − 5.36e12·33-s − 3.40e12·35-s − 1.98e13·37-s + 1.96e12·39-s + 1.46e13·41-s − 1.16e14·43-s − 7.04e12·45-s + 1.76e14·47-s + 2.01e14·49-s − 2.93e14·51-s + 1.52e14·53-s + 1.33e14·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.187·5-s + 1.36·7-s + 1/3·9-s − 1.14·11-s + 0.101·13-s − 0.108·15-s − 1.55·17-s − 1.06·19-s + 0.789·21-s + 1.87·23-s − 0.964·25-s + 0.192·27-s − 0.0608·29-s − 0.221·31-s − 0.663·33-s − 0.255·35-s − 0.926·37-s + 0.0588·39-s + 0.286·41-s − 1.51·43-s − 0.0624·45-s + 1.08·47-s + 0.868·49-s − 0.898·51-s + 0.337·53-s + 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p^{8} T \) |
| good | 5 | \( 1 + 163554 T + p^{17} T^{2} \) |
| 7 | \( 1 - 425440 p^{2} T + p^{17} T^{2} \) |
| 11 | \( 1 + 817372356 T + p^{17} T^{2} \) |
| 13 | \( 1 - 23045366 p T + p^{17} T^{2} \) |
| 17 | \( 1 + 44775606078 T + p^{17} T^{2} \) |
| 19 | \( 1 + 78748651964 T + p^{17} T^{2} \) |
| 23 | \( 1 - 704672009160 T + p^{17} T^{2} \) |
| 29 | \( 1 + 163793785242 T + p^{17} T^{2} \) |
| 31 | \( 1 + 1049860831400 T + p^{17} T^{2} \) |
| 37 | \( 1 + 19805735857210 T + p^{17} T^{2} \) |
| 41 | \( 1 - 14660035932090 T + p^{17} T^{2} \) |
| 43 | \( 1 + 116038864682564 T + p^{17} T^{2} \) |
| 47 | \( 1 - 176606594594112 T + p^{17} T^{2} \) |
| 53 | \( 1 - 152863496635230 T + p^{17} T^{2} \) |
| 59 | \( 1 - 262797291296124 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1358552281482562 T + p^{17} T^{2} \) |
| 67 | \( 1 + 444863620615292 T + p^{17} T^{2} \) |
| 71 | \( 1 - 4003270764790968 T + p^{17} T^{2} \) |
| 73 | \( 1 - 924832535317130 T + p^{17} T^{2} \) |
| 79 | \( 1 + 14747307742797080 T + p^{17} T^{2} \) |
| 83 | \( 1 + 26422963268810172 T + p^{17} T^{2} \) |
| 89 | \( 1 + 38883748080645126 T + p^{17} T^{2} \) |
| 97 | \( 1 + 25374394856250238 T + p^{17} T^{2} \) |
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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
−11.32926339483617596510351985381, −10.57386163956959805240358692803, −8.891039716065014652243471957358, −8.133117985402264084110052831863, −6.99791215759542066567671013605, −5.20528760711523407858837732756, −4.23717398529690426936416787312, −2.62222131009178118346420869279, −1.63902272712204523903605356482, 0,
1.63902272712204523903605356482, 2.62222131009178118346420869279, 4.23717398529690426936416787312, 5.20528760711523407858837732756, 6.99791215759542066567671013605, 8.133117985402264084110052831863, 8.891039716065014652243471957358, 10.57386163956959805240358692803, 11.32926339483617596510351985381
