| L(s) = 1 | − 2-s + 4-s − 8-s + 3·11-s − 2·13-s + 16-s − 3·22-s − 12·23-s − 7·25-s + 2·26-s − 32-s + 10·37-s + 3·44-s + 12·46-s − 6·47-s + 5·49-s + 7·50-s − 2·52-s − 12·59-s + 4·61-s + 64-s + 73-s − 10·74-s − 6·83-s − 3·88-s − 12·92-s + 6·94-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.904·11-s − 0.554·13-s + 1/4·16-s − 0.639·22-s − 2.50·23-s − 7/5·25-s + 0.392·26-s − 0.176·32-s + 1.64·37-s + 0.452·44-s + 1.76·46-s − 0.875·47-s + 5/7·49-s + 0.989·50-s − 0.277·52-s − 1.56·59-s + 0.512·61-s + 1/8·64-s + 0.117·73-s − 1.16·74-s − 0.658·83-s − 0.319·88-s − 1.25·92-s + 0.618·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.873406254542253615532518510699, −8.186233180787386844183617259045, −8.006366778069969960041194579070, −7.53300008675989380577093243961, −6.99776914128901841444220230353, −6.30849189421461885942335153444, −6.07951319579355213241880943639, −5.58056742219958466607894336935, −4.74043794922922414446861353346, −4.03677392530382822664018283403, −3.81478314028392958267546325330, −2.77697357232355341408669209477, −2.13461549641041127002307140108, −1.41864583593521888139090221982, 0,
1.41864583593521888139090221982, 2.13461549641041127002307140108, 2.77697357232355341408669209477, 3.81478314028392958267546325330, 4.03677392530382822664018283403, 4.74043794922922414446861353346, 5.58056742219958466607894336935, 6.07951319579355213241880943639, 6.30849189421461885942335153444, 6.99776914128901841444220230353, 7.53300008675989380577093243961, 8.006366778069969960041194579070, 8.186233180787386844183617259045, 8.873406254542253615532518510699
