L(s) = 1 | − 4·2-s − 11·3-s + 16·4-s + 44·6-s − 142·7-s − 64·8-s − 122·9-s + 777·11-s − 176·12-s + 884·13-s + 568·14-s + 256·16-s − 27·17-s + 488·18-s + 1.14e3·19-s + 1.56e3·21-s − 3.10e3·22-s + 1.85e3·23-s + 704·24-s − 3.53e3·26-s + 4.01e3·27-s − 2.27e3·28-s − 4.92e3·29-s + 1.80e3·31-s − 1.02e3·32-s − 8.54e3·33-s + 108·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.705·3-s + 1/2·4-s + 0.498·6-s − 1.09·7-s − 0.353·8-s − 0.502·9-s + 1.93·11-s − 0.352·12-s + 1.45·13-s + 0.774·14-s + 1/4·16-s − 0.0226·17-s + 0.355·18-s + 0.727·19-s + 0.772·21-s − 1.36·22-s + 0.730·23-s + 0.249·24-s − 1.02·26-s + 1.05·27-s − 0.547·28-s − 1.08·29-s + 0.336·31-s − 0.176·32-s − 1.36·33-s + 0.0160·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8773000565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8773000565\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 11 T + p^{5} T^{2} \) |
| 7 | \( 1 + 142 T + p^{5} T^{2} \) |
| 11 | \( 1 - 777 T + p^{5} T^{2} \) |
| 13 | \( 1 - 68 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 27 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1145 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1854 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4920 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1802 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13178 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15123 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7844 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6732 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3414 T + p^{5} T^{2} \) |
| 59 | \( 1 - 33960 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47402 T + p^{5} T^{2} \) |
| 67 | \( 1 + 13177 T + p^{5} T^{2} \) |
| 71 | \( 1 + 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 + 59821 T + p^{5} T^{2} \) |
| 79 | \( 1 - 75830 T + p^{5} T^{2} \) |
| 83 | \( 1 - 46299 T + p^{5} T^{2} \) |
| 89 | \( 1 + 30585 T + p^{5} T^{2} \) |
| 97 | \( 1 - 104018 T + p^{5} T^{2} \) |
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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
−14.69954900633123409744353334037, −13.27444861941137286625604157354, −11.83387151628347309331008611882, −11.14347824453881640232105246972, −9.637245151234445549359035256651, −8.720014688138801704735431270894, −6.76808591828028461609603297738, −5.96933655137361665115103951023, −3.52789039826596640252118771005, −0.945162218597550160532026810049,
0.945162218597550160532026810049, 3.52789039826596640252118771005, 5.96933655137361665115103951023, 6.76808591828028461609603297738, 8.720014688138801704735431270894, 9.637245151234445549359035256651, 11.14347824453881640232105246972, 11.83387151628347309331008611882, 13.27444861941137286625604157354, 14.69954900633123409744353334037