L(s) = 1 | + 7-s + 4·9-s + 2·11-s − 2·23-s − 2·25-s − 6·29-s − 14·37-s − 6·43-s + 49-s − 8·53-s + 4·63-s − 4·67-s − 4·71-s + 2·77-s + 7·81-s + 8·99-s − 20·107-s + 2·109-s + 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 4/3·9-s + 0.603·11-s − 0.417·23-s − 2/5·25-s − 1.11·29-s − 2.30·37-s − 0.914·43-s + 1/7·49-s − 1.09·53-s + 0.503·63-s − 0.488·67-s − 0.474·71-s + 0.227·77-s + 7/9·81-s + 0.804·99-s − 1.93·107-s + 0.191·109-s + 1.50·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−7.69482405895834397367412039494, −7.30643370909968080566606447899, −6.82759137091116655199567209076, −6.59719074620483121754814490161, −6.00609131104675387706915268936, −5.45987015954679960643740917137, −5.06574236903714323014596414504, −4.54542721719265047362564008627, −4.08256113897269383538103799877, −3.65194203381077251607794215944, −3.21241631286068701200752599073, −2.27372078903964464441170712081, −1.66066384870243542860380705788, −1.36585965362579962812780371947, 0,
1.36585965362579962812780371947, 1.66066384870243542860380705788, 2.27372078903964464441170712081, 3.21241631286068701200752599073, 3.65194203381077251607794215944, 4.08256113897269383538103799877, 4.54542721719265047362564008627, 5.06574236903714323014596414504, 5.45987015954679960643740917137, 6.00609131104675387706915268936, 6.59719074620483121754814490161, 6.82759137091116655199567209076, 7.30643370909968080566606447899, 7.69482405895834397367412039494