| L(s) = 1 | − 3-s + 4-s + 6·7-s − 2·9-s − 12-s − 2·13-s + 16-s − 6·21-s − 9·25-s + 5·27-s + 6·28-s + 4·31-s − 2·36-s − 4·37-s + 2·39-s + 8·43-s − 48-s + 13·49-s − 2·52-s + 24·61-s − 12·63-s + 64-s + 16·67-s − 12·73-s + 9·75-s − 2·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 2.26·7-s − 2/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.30·21-s − 9/5·25-s + 0.962·27-s + 1.13·28-s + 0.718·31-s − 1/3·36-s − 0.657·37-s + 0.320·39-s + 1.21·43-s − 0.144·48-s + 13/7·49-s − 0.277·52-s + 3.07·61-s − 1.51·63-s + 1/8·64-s + 1.95·67-s − 1.40·73-s + 1.03·75-s − 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881765723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881765723\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.747141121901939638246702173369, −8.520482678541895661066213736227, −8.054137528537468354465526888801, −7.50453196645377754362608030446, −7.39141785314113770727760411934, −6.47484848767999948735932618155, −6.11586748167354289065343430906, −5.34006059048988750874758197539, −5.27606039091750975262211940149, −4.62642136644695957414242142502, −4.08090613362146692342747948187, −3.29745348810458593262782421415, −2.19477689281745387953094778021, −2.06983106012671431074676520675, −0.905850257464074624516301543141,
0.905850257464074624516301543141, 2.06983106012671431074676520675, 2.19477689281745387953094778021, 3.29745348810458593262782421415, 4.08090613362146692342747948187, 4.62642136644695957414242142502, 5.27606039091750975262211940149, 5.34006059048988750874758197539, 6.11586748167354289065343430906, 6.47484848767999948735932618155, 7.39141785314113770727760411934, 7.50453196645377754362608030446, 8.054137528537468354465526888801, 8.520482678541895661066213736227, 8.747141121901939638246702173369
