| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 2·11-s − 12-s − 13-s + 2·14-s + 16-s + 17-s − 18-s + 6·19-s + 2·21-s + 2·22-s − 6·23-s + 24-s + 26-s − 27-s − 2·28-s − 8·29-s + 7·31-s − 32-s + 2·33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.436·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.377·28-s − 1.48·29-s + 1.25·31-s − 0.176·32-s + 0.348·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2867589461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2867589461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.09037728211261, −12.23852657889869, −12.18228116059934, −11.58722169796213, −11.23306846530000, −10.42737862392250, −10.27001667396192, −9.832266312386402, −9.295803104336561, −9.002245977017825, −8.142979923137745, −7.725209899148456, −7.461233901298792, −6.832688353601775, −6.217912335079281, −5.953313135512958, −5.337494346421068, −4.853669576957340, −4.179371407308215, −3.375128347256123, −3.134214543723580, −2.330127904388571, −1.714869911807502, −1.030293879922288, −0.1963645158728518,
0.1963645158728518, 1.030293879922288, 1.714869911807502, 2.330127904388571, 3.134214543723580, 3.375128347256123, 4.179371407308215, 4.853669576957340, 5.337494346421068, 5.953313135512958, 6.217912335079281, 6.832688353601775, 7.461233901298792, 7.725209899148456, 8.142979923137745, 9.002245977017825, 9.295803104336561, 9.832266312386402, 10.27001667396192, 10.42737862392250, 11.23306846530000, 11.58722169796213, 12.18228116059934, 12.23852657889869, 13.09037728211261
