L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s − 4·13-s + 16-s + 4·17-s + 18-s + 2·20-s + 3·25-s − 4·26-s + 12·29-s + 32-s + 4·34-s + 36-s − 4·37-s + 2·40-s + 4·41-s + 2·45-s + 49-s + 3·50-s − 4·52-s + 12·53-s + 12·58-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s − 0.784·26-s + 2.22·29-s + 0.176·32-s + 0.685·34-s + 1/6·36-s − 0.657·37-s + 0.316·40-s + 0.624·41-s + 0.298·45-s + 1/7·49-s + 0.424·50-s − 0.554·52-s + 1.64·53-s + 1.57·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.665899970\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.665899970\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.703442590199731071990143235117, −8.287096973110017361623722641321, −7.62519571288793976021047667530, −7.22506991546840018830721628191, −6.91101846984498094297070698671, −6.07862262147452622495562430171, −6.07443290941258095916590657527, −5.23689790182682732993652607626, −4.97432253089418114386617517257, −4.42770208936219550864886452429, −3.82546629697594474148073649640, −2.94489120047115936245774390265, −2.68744299862141036032959354712, −1.87682589231908259904942053412, −1.03754454711477347129114045578,
1.03754454711477347129114045578, 1.87682589231908259904942053412, 2.68744299862141036032959354712, 2.94489120047115936245774390265, 3.82546629697594474148073649640, 4.42770208936219550864886452429, 4.97432253089418114386617517257, 5.23689790182682732993652607626, 6.07443290941258095916590657527, 6.07862262147452622495562430171, 6.91101846984498094297070698671, 7.22506991546840018830721628191, 7.62519571288793976021047667530, 8.287096973110017361623722641321, 8.703442590199731071990143235117