| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 7-s + 3·8-s + 9-s + 10-s + 5·11-s − 12-s + 13-s + 14-s − 15-s − 16-s − 18-s + 5·19-s + 20-s − 21-s − 5·22-s + 3·24-s + 25-s − 26-s + 27-s + 28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.218·21-s − 1.06·22-s + 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.56750675048205, −14.29029512668825, −13.61272196117133, −13.18786822700562, −12.75406667300614, −12.04455888768001, −11.47530187256024, −11.16365908012388, −10.34742419288102, −9.766210927772739, −9.379222119647661, −9.147295644616712, −8.376107155622309, −8.076540647081647, −7.457036468124821, −6.843597536619766, −6.477989100018564, −5.492930487822429, −4.981360089635914, −4.198511134362858, −3.593250292952684, −3.474732279953622, −2.327042817754098, −1.469447703792904, −1.002478769379025, 0,
1.002478769379025, 1.469447703792904, 2.327042817754098, 3.474732279953622, 3.593250292952684, 4.198511134362858, 4.981360089635914, 5.492930487822429, 6.477989100018564, 6.843597536619766, 7.457036468124821, 8.076540647081647, 8.376107155622309, 9.147295644616712, 9.379222119647661, 9.766210927772739, 10.34742419288102, 11.16365908012388, 11.47530187256024, 12.04455888768001, 12.75406667300614, 13.18786822700562, 13.61272196117133, 14.29029512668825, 14.56750675048205
