| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 216·6-s − 349·7-s − 512·8-s + 729·9-s + 1.18e3·11-s − 1.72e3·12-s − 1.72e3·13-s + 2.79e3·14-s + 4.09e3·16-s − 7.49e3·17-s − 5.83e3·18-s + 1.27e4·19-s + 9.42e3·21-s − 9.45e3·22-s + 6.40e3·23-s + 1.38e4·24-s + 1.37e4·26-s − 1.96e4·27-s − 2.23e4·28-s + 1.08e5·29-s + 1.42e5·31-s − 3.27e4·32-s − 3.19e4·33-s + 5.99e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.384·7-s − 0.353·8-s + 1/3·9-s + 0.267·11-s − 0.288·12-s − 0.217·13-s + 0.271·14-s + 1/4·16-s − 0.369·17-s − 0.235·18-s + 0.427·19-s + 0.222·21-s − 0.189·22-s + 0.109·23-s + 0.204·24-s + 0.153·26-s − 0.192·27-s − 0.192·28-s + 0.822·29-s + 0.858·31-s − 0.176·32-s − 0.154·33-s + 0.261·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 349 T + p^{7} T^{2} \) |
| 11 | \( 1 - 1182 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1723 T + p^{7} T^{2} \) |
| 17 | \( 1 + 7494 T + p^{7} T^{2} \) |
| 19 | \( 1 - 12785 T + p^{7} T^{2} \) |
| 23 | \( 1 - 6402 T + p^{7} T^{2} \) |
| 29 | \( 1 - 108090 T + p^{7} T^{2} \) |
| 31 | \( 1 - 142427 T + p^{7} T^{2} \) |
| 37 | \( 1 - 276266 T + p^{7} T^{2} \) |
| 41 | \( 1 - 525072 T + p^{7} T^{2} \) |
| 43 | \( 1 + 747013 T + p^{7} T^{2} \) |
| 47 | \( 1 - 571326 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1472028 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1582110 T + p^{7} T^{2} \) |
| 61 | \( 1 + 932893 T + p^{7} T^{2} \) |
| 67 | \( 1 + 1688089 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2962752 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4078798 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5635360 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3120318 T + p^{7} T^{2} \) |
| 89 | \( 1 + 9155040 T + p^{7} T^{2} \) |
| 97 | \( 1 + 10041199 T + p^{7} T^{2} \) |
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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
−11.12694845589798813694123368819, −10.09880566160050835078353020577, −9.276093177671324661450215698427, −8.043075379459003587699181091881, −6.88886532025233144731988666985, −5.98949402032303233485521144938, −4.54052090392755121003541752833, −2.88641641666643973884806475781, −1.27871572286614365546431515389, 0,
1.27871572286614365546431515389, 2.88641641666643973884806475781, 4.54052090392755121003541752833, 5.98949402032303233485521144938, 6.88886532025233144731988666985, 8.043075379459003587699181091881, 9.276093177671324661450215698427, 10.09880566160050835078353020577, 11.12694845589798813694123368819
