| L(s) = 1 | − 128·2-s + 1.63e4·4-s − 7.76e4·5-s + 7.62e5·7-s − 2.09e6·8-s + 9.93e6·10-s − 4.80e7·11-s + 2.85e8·13-s − 9.75e7·14-s + 2.68e8·16-s + 3.17e9·17-s − 5.89e9·19-s − 1.27e9·20-s + 6.14e9·22-s + 3.33e8·23-s − 2.44e10·25-s − 3.64e10·26-s + 1.24e10·28-s − 1.17e11·29-s − 2.25e11·31-s − 3.43e10·32-s − 4.06e11·34-s − 5.91e10·35-s − 4.77e11·37-s + 7.54e11·38-s + 1.62e11·40-s − 1.19e12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.444·5-s + 0.349·7-s − 0.353·8-s + 0.314·10-s − 0.742·11-s + 1.26·13-s − 0.247·14-s + 1/4·16-s + 1.87·17-s − 1.51·19-s − 0.222·20-s + 0.525·22-s + 0.0203·23-s − 0.802·25-s − 0.891·26-s + 0.174·28-s − 1.26·29-s − 1.47·31-s − 0.176·32-s − 1.32·34-s − 0.155·35-s − 0.827·37-s + 1.06·38-s + 0.157·40-s − 0.959·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 77646 T + p^{15} T^{2} \) |
| 7 | \( 1 - 108872 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 4364652 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 21933086 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 3173671566 T + p^{15} T^{2} \) |
| 19 | \( 1 + 5895116260 T + p^{15} T^{2} \) |
| 23 | \( 1 - 333010392 T + p^{15} T^{2} \) |
| 29 | \( 1 + 117285392310 T + p^{15} T^{2} \) |
| 31 | \( 1 + 225821452768 T + p^{15} T^{2} \) |
| 37 | \( 1 + 477657973906 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1196721561882 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1066802913668 T + p^{15} T^{2} \) |
| 47 | \( 1 + 1324913565264 T + p^{15} T^{2} \) |
| 53 | \( 1 - 6573181204962 T + p^{15} T^{2} \) |
| 59 | \( 1 + 7973946241140 T + p^{15} T^{2} \) |
| 61 | \( 1 - 14311350203222 T + p^{15} T^{2} \) |
| 67 | \( 1 - 41052380998124 T + p^{15} T^{2} \) |
| 71 | \( 1 + 67253761134072 T + p^{15} T^{2} \) |
| 73 | \( 1 + 156200366359942 T + p^{15} T^{2} \) |
| 79 | \( 1 + 138004701018640 T + p^{15} T^{2} \) |
| 83 | \( 1 + 469396029824988 T + p^{15} T^{2} \) |
| 89 | \( 1 - 422649074576790 T + p^{15} T^{2} \) |
| 97 | \( 1 + 201862519502686 T + p^{15} T^{2} \) |
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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
−14.65200872019784430416510463432, −12.90090046003156831724174452613, −11.41427181348589689776052072905, −10.29265357452744372394710257261, −8.603696492439207045197001822510, −7.54834229168588260930156614407, −5.71874440517255125199592008218, −3.60144223120983576285641465620, −1.65677026376851901593905306185, 0,
1.65677026376851901593905306185, 3.60144223120983576285641465620, 5.71874440517255125199592008218, 7.54834229168588260930156614407, 8.603696492439207045197001822510, 10.29265357452744372394710257261, 11.41427181348589689776052072905, 12.90090046003156831724174452613, 14.65200872019784430416510463432
