L(s) = 1 | − 4·5-s + 7-s − 6·9-s − 8·11-s − 4·13-s + 2·25-s − 16·31-s − 4·35-s − 8·43-s + 24·45-s + 16·47-s + 49-s + 32·55-s + 12·61-s − 6·63-s + 16·65-s − 8·67-s − 8·77-s + 27·81-s − 4·91-s + 48·99-s − 4·101-s + 32·103-s − 24·107-s + 4·113-s + 24·117-s + 26·121-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 0.377·7-s − 2·9-s − 2.41·11-s − 1.10·13-s + 2/5·25-s − 2.87·31-s − 0.676·35-s − 1.21·43-s + 3.57·45-s + 2.33·47-s + 1/7·49-s + 4.31·55-s + 1.53·61-s − 0.755·63-s + 1.98·65-s − 0.977·67-s − 0.911·77-s + 3·81-s − 0.419·91-s + 4.82·99-s − 0.398·101-s + 3.15·103-s − 2.32·107-s + 0.376·113-s + 2.21·117-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.749304120934004278681759852831, −8.078246359041317001571436271675, −7.79292083571821938290976650763, −7.30264187759595822059178656689, −7.26382021666693896035142333427, −5.92791093661634117809390012383, −5.69394971603405508863657314834, −4.98409920426632935260415692638, −4.92225264716551837532790929786, −3.68340574560716411313233602487, −3.60173413237973468362473708205, −2.50561452898491372410349134801, −2.40145156924384381758797773878, 0, 0,
2.40145156924384381758797773878, 2.50561452898491372410349134801, 3.60173413237973468362473708205, 3.68340574560716411313233602487, 4.92225264716551837532790929786, 4.98409920426632935260415692638, 5.69394971603405508863657314834, 5.92791093661634117809390012383, 7.26382021666693896035142333427, 7.30264187759595822059178656689, 7.79292083571821938290976650763, 8.078246359041317001571436271675, 8.749304120934004278681759852831