| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 11-s − 12-s − 13-s − 2·14-s + 16-s − 6·17-s + 18-s − 4·19-s + 2·21-s − 22-s − 24-s − 26-s − 27-s − 2·28-s + 2·31-s + 32-s + 33-s − 6·34-s + 36-s + 4·37-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 − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.213·22-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.359·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319909463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319909463\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−15.54797836036230, −15.10557290944980, −14.55166122235701, −13.77359027967474, −13.29910090484821, −12.85363864358873, −12.45724344355023, −11.80938915801063, −11.15450281848289, −10.80641730193298, −10.18936948983135, −9.531316852914139, −8.982927195909700, −8.162666930999097, −7.539292399723613, −6.797857319994641, −6.262636838140787, −6.073382738495750, −4.940629572267220, −4.687872413982345, −3.947018804125664, −3.159214022171171, −2.441418708386391, −1.712788387393657, −0.4098198380380412,
0.4098198380380412, 1.712788387393657, 2.441418708386391, 3.159214022171171, 3.947018804125664, 4.687872413982345, 4.940629572267220, 6.073382738495750, 6.262636838140787, 6.797857319994641, 7.539292399723613, 8.162666930999097, 8.982927195909700, 9.531316852914139, 10.18936948983135, 10.80641730193298, 11.15450281848289, 11.80938915801063, 12.45724344355023, 12.85363864358873, 13.29910090484821, 13.77359027967474, 14.55166122235701, 15.10557290944980, 15.54797836036230
