| L(s) = 1 | − 6.25e3·3-s + 9.05e4·5-s − 56·7-s + 2.47e7·9-s + 9.58e7·11-s − 5.97e7·13-s − 5.65e8·15-s − 1.35e9·17-s − 3.78e9·19-s + 3.50e5·21-s + 1.16e10·23-s − 2.23e10·25-s − 6.49e10·27-s − 2.89e10·29-s − 2.53e11·31-s − 5.99e11·33-s − 5.06e6·35-s + 8.17e11·37-s + 3.73e11·39-s − 6.82e11·41-s − 3.66e11·43-s + 2.23e12·45-s − 6.95e11·47-s − 4.74e12·49-s + 8.47e12·51-s + 1.29e13·53-s + 8.67e12·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s + 0.518·5-s − 2.57e − 5·7-s + 1.72·9-s + 1.48·11-s − 0.264·13-s − 0.855·15-s − 0.801·17-s − 0.971·19-s + 4.24e−5·21-s + 0.710·23-s − 0.731·25-s − 1.19·27-s − 0.311·29-s − 1.65·31-s − 2.44·33-s − 1.33e − 5·35-s + 1.41·37-s + 0.436·39-s − 0.547·41-s − 0.205·43-s + 0.893·45-s − 0.200·47-s − 0.999·49-s + 1.32·51-s + 1.51·53-s + 0.768·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 2084 p T + p^{15} T^{2} \) |
| 5 | \( 1 - 18102 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 8 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 8717268 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 4598626 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1355814414 T + p^{15} T^{2} \) |
| 19 | \( 1 + 3783593180 T + p^{15} T^{2} \) |
| 23 | \( 1 - 11608845528 T + p^{15} T^{2} \) |
| 29 | \( 1 + 28959105930 T + p^{15} T^{2} \) |
| 31 | \( 1 + 253685353952 T + p^{15} T^{2} \) |
| 37 | \( 1 - 817641294446 T + p^{15} T^{2} \) |
| 41 | \( 1 + 682333284198 T + p^{15} T^{2} \) |
| 43 | \( 1 + 366945604292 T + p^{15} T^{2} \) |
| 47 | \( 1 + 695741581776 T + p^{15} T^{2} \) |
| 53 | \( 1 - 12993372468702 T + p^{15} T^{2} \) |
| 59 | \( 1 + 9209035340340 T + p^{15} T^{2} \) |
| 61 | \( 1 + 42338641200298 T + p^{15} T^{2} \) |
| 67 | \( 1 + 448205790308 p T + p^{15} T^{2} \) |
| 71 | \( 1 + 115328696975352 T + p^{15} T^{2} \) |
| 73 | \( 1 - 43787346432122 T + p^{15} T^{2} \) |
| 79 | \( 1 + 79603813043120 T + p^{15} T^{2} \) |
| 83 | \( 1 - 41169504396 p T + p^{15} T^{2} \) |
| 89 | \( 1 + 377306179184790 T + p^{15} T^{2} \) |
| 97 | \( 1 + 166982186657374 T + p^{15} T^{2} \) |
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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
−14.88051941984458939556439170569, −13.08924517360831354021443392888, −11.81455068259714937575507358379, −10.82041375557016898963536667404, −9.319839354321568294909832183196, −6.83589187619535710256837679404, −5.84631668552315150872505679450, −4.34500764651545938710099320779, −1.56785227641473819088012327695, 0,
1.56785227641473819088012327695, 4.34500764651545938710099320779, 5.84631668552315150872505679450, 6.83589187619535710256837679404, 9.319839354321568294909832183196, 10.82041375557016898963536667404, 11.81455068259714937575507358379, 13.08924517360831354021443392888, 14.88051941984458939556439170569
