| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 216·6-s + 713·7-s − 512·8-s + 729·9-s + 3.81e3·11-s + 1.72e3·12-s − 391·13-s − 5.70e3·14-s + 4.09e3·16-s − 4.18e3·17-s − 5.83e3·18-s − 1.56e3·19-s + 1.92e4·21-s − 3.04e4·22-s + 1.14e5·23-s − 1.38e4·24-s + 3.12e3·26-s + 1.96e4·27-s + 4.56e4·28-s − 8.32e4·29-s − 8.31e4·31-s − 3.27e4·32-s + 1.02e5·33-s + 3.34e4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.785·7-s − 0.353·8-s + 1/3·9-s + 0.863·11-s + 0.288·12-s − 0.0493·13-s − 0.555·14-s + 1/4·16-s − 0.206·17-s − 0.235·18-s − 0.0522·19-s + 0.453·21-s − 0.610·22-s + 1.95·23-s − 0.204·24-s + 0.0349·26-s + 0.192·27-s + 0.392·28-s − 0.633·29-s − 0.501·31-s − 0.176·32-s + 0.498·33-s + 0.145·34-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
| \(L(4)\) |
\(\approx\) |
\(2.223088830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223088830\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 713 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3810 T + p^{7} T^{2} \) |
| 13 | \( 1 + 391 T + p^{7} T^{2} \) |
| 17 | \( 1 + 246 p T + p^{7} T^{2} \) |
| 19 | \( 1 + 1561 T + p^{7} T^{2} \) |
| 23 | \( 1 - 114150 T + p^{7} T^{2} \) |
| 29 | \( 1 + 83214 T + p^{7} T^{2} \) |
| 31 | \( 1 + 83167 T + p^{7} T^{2} \) |
| 37 | \( 1 + 231334 T + p^{7} T^{2} \) |
| 41 | \( 1 + 124656 T + p^{7} T^{2} \) |
| 43 | \( 1 - 193757 T + p^{7} T^{2} \) |
| 47 | \( 1 - 319290 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1645428 T + p^{7} T^{2} \) |
| 59 | \( 1 + 38610 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1973905 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4409753 T + p^{7} T^{2} \) |
| 71 | \( 1 - 124080 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3967634 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7107992 T + p^{7} T^{2} \) |
| 83 | \( 1 - 8117694 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6727872 T + p^{7} T^{2} \) |
| 97 | \( 1 + 14268679 T + p^{7} T^{2} \) |
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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
−11.45921194648592781667234084880, −10.66793854975150206855935899270, −9.339478086748993728805354471289, −8.740987090324215724318964116006, −7.62477996435617221779927777086, −6.68807695357198084146737698173, −5.06826054908479792590959942658, −3.59148035168922448000544851468, −2.11247490149974123584653563129, −0.970133117928985016681601604619,
0.970133117928985016681601604619, 2.11247490149974123584653563129, 3.59148035168922448000544851468, 5.06826054908479792590959942658, 6.68807695357198084146737698173, 7.62477996435617221779927777086, 8.740987090324215724318964116006, 9.339478086748993728805354471289, 10.66793854975150206855935899270, 11.45921194648592781667234084880
