| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s + 9-s − 8·11-s − 2·14-s + 5·16-s + 2·18-s − 16·22-s − 16·23-s + 25-s − 3·28-s − 4·29-s + 6·32-s + 3·36-s + 12·37-s − 8·43-s − 24·44-s − 32·46-s + 49-s + 2·50-s − 20·53-s − 4·56-s − 8·58-s − 63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s + 1/3·9-s − 2.41·11-s − 0.534·14-s + 5/4·16-s + 0.471·18-s − 3.41·22-s − 3.33·23-s + 1/5·25-s − 0.566·28-s − 0.742·29-s + 1.06·32-s + 1/2·36-s + 1.97·37-s − 1.21·43-s − 3.61·44-s − 4.71·46-s + 1/7·49-s + 0.282·50-s − 2.74·53-s − 0.534·56-s − 1.05·58-s − 0.125·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.185634457079877915713969006222, −7.987738483103106114928045639009, −7.56039077916841084023228832876, −7.31970008287706573052341981271, −6.28350214525854496920877449906, −5.97975998469911468351052676550, −5.94316092695460714490045547798, −4.95725936124254114017607791343, −4.76181305505936911086603979238, −4.20752575410863341791747093067, −3.44893990823261081236809071688, −3.05810466361682329972112745562, −2.27051973415844153304852765225, −1.88155902079073536734325978904, 0,
1.88155902079073536734325978904, 2.27051973415844153304852765225, 3.05810466361682329972112745562, 3.44893990823261081236809071688, 4.20752575410863341791747093067, 4.76181305505936911086603979238, 4.95725936124254114017607791343, 5.94316092695460714490045547798, 5.97975998469911468351052676550, 6.28350214525854496920877449906, 7.31970008287706573052341981271, 7.56039077916841084023228832876, 7.987738483103106114928045639009, 8.185634457079877915713969006222
