| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 4·7-s + 3·8-s + 9-s − 10-s − 12-s + 6·13-s + 4·14-s + 15-s − 16-s + 2·17-s − 18-s + 8·19-s − 20-s − 4·21-s − 4·23-s + 3·24-s + 25-s − 6·26-s + 27-s + 4·28-s + 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 − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.872·21-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045386567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045386567\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.75800573511653481896464129178, −9.760218247853831386687609022861, −9.557503701967675291305833337136, −8.575570814640394375498951194613, −7.72786156373545594249219051552, −6.55543097381440476299846924590, −5.58153071011565013738124935193, −3.97599829860772318843336935157, −3.06133580739474070245483773115, −1.13897079254288435205597284658,
1.13897079254288435205597284658, 3.06133580739474070245483773115, 3.97599829860772318843336935157, 5.58153071011565013738124935193, 6.55543097381440476299846924590, 7.72786156373545594249219051552, 8.575570814640394375498951194613, 9.557503701967675291305833337136, 9.760218247853831386687609022861, 10.75800573511653481896464129178
