| L(s) = 1 | − 4·4-s − 3·7-s + 9-s − 6·11-s + 12·16-s + 6·23-s − 9·25-s + 12·28-s − 2·29-s − 4·36-s + 2·37-s − 16·43-s + 24·44-s + 2·49-s + 20·53-s − 3·63-s − 32·64-s + 8·67-s − 12·71-s + 18·77-s − 6·79-s + 81-s − 24·92-s − 6·99-s + 36·100-s + 18·107-s − 36·112-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.13·7-s + 1/3·9-s − 1.80·11-s + 3·16-s + 1.25·23-s − 9/5·25-s + 2.26·28-s − 0.371·29-s − 2/3·36-s + 0.328·37-s − 2.43·43-s + 3.61·44-s + 2/7·49-s + 2.74·53-s − 0.377·63-s − 4·64-s + 0.977·67-s − 1.42·71-s + 2.05·77-s − 0.675·79-s + 1/9·81-s − 2.50·92-s − 0.603·99-s + 18/5·100-s + 1.74·107-s − 3.40·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 974169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 974169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.100368548472543721258158069369, −7.40920256558116385528575072059, −7.30736849693508971465380420611, −6.58166556711700165328524737731, −5.83486052932247599921983260982, −5.65547826783607225275216941973, −5.17586605015791263421626852895, −4.75349830419579613526576050031, −4.26605262750633605937690582473, −3.66487809250971104445799041503, −3.31847810114563157193588872502, −2.76252878185140238150448923503, −1.85889915579708054803892223778, −0.69581504422538865690992345995, 0,
0.69581504422538865690992345995, 1.85889915579708054803892223778, 2.76252878185140238150448923503, 3.31847810114563157193588872502, 3.66487809250971104445799041503, 4.26605262750633605937690582473, 4.75349830419579613526576050031, 5.17586605015791263421626852895, 5.65547826783607225275216941973, 5.83486052932247599921983260982, 6.58166556711700165328524737731, 7.30736849693508971465380420611, 7.40920256558116385528575072059, 8.100368548472543721258158069369
