L(s) = 1 | − 3·4-s − 3·9-s + 4·13-s + 5·16-s − 8·19-s + 25-s − 16·31-s + 9·36-s − 4·37-s + 8·43-s − 14·49-s − 12·52-s − 20·61-s − 3·64-s − 32·67-s + 28·73-s + 24·76-s + 16·79-s + 9·81-s + 20·97-s − 3·100-s − 8·103-s − 36·109-s − 12·117-s + 121-s + 48·124-s + 127-s + ⋯ |
L(s) = 1 | − 3/2·4-s − 9-s + 1.10·13-s + 5/4·16-s − 1.83·19-s + 1/5·25-s − 2.87·31-s + 3/2·36-s − 0.657·37-s + 1.21·43-s − 2·49-s − 1.66·52-s − 2.56·61-s − 3/8·64-s − 3.90·67-s + 3.27·73-s + 2.75·76-s + 1.80·79-s + 81-s + 2.03·97-s − 0.299·100-s − 0.788·103-s − 3.44·109-s − 1.10·117-s + 1/11·121-s + 4.31·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−10.63828874077951155959665793654, −9.365267708430525132762162914792, −9.301659499748329006626376523886, −8.877176431244694382657420951838, −8.248642768475844318802620430882, −7.963920428420679646091651044326, −7.06776652987294055072227501878, −6.16821463973303244938343944630, −5.89992194836808220350577312387, −5.09766059609940450216711100862, −4.52659700416500112028212971426, −3.79000489498233733500749844862, −3.25423179226253980082861279896, −1.86919846930486895513918273600, 0,
1.86919846930486895513918273600, 3.25423179226253980082861279896, 3.79000489498233733500749844862, 4.52659700416500112028212971426, 5.09766059609940450216711100862, 5.89992194836808220350577312387, 6.16821463973303244938343944630, 7.06776652987294055072227501878, 7.963920428420679646091651044326, 8.248642768475844318802620430882, 8.877176431244694382657420951838, 9.301659499748329006626376523886, 9.365267708430525132762162914792, 10.63828874077951155959665793654