| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 2·5-s − 4·6-s + 2·9-s − 4·10-s + 2·11-s − 4·12-s + 6·13-s + 4·15-s − 4·16-s + 4·18-s + 6·19-s − 4·20-s + 4·22-s + 2·25-s + 12·26-s − 6·27-s + 4·29-s + 8·30-s − 8·32-s − 4·33-s + 4·36-s − 14·37-s + 12·38-s − 12·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 2/3·9-s − 1.26·10-s + 0.603·11-s − 1.15·12-s + 1.66·13-s + 1.03·15-s − 16-s + 0.942·18-s + 1.37·19-s − 0.894·20-s + 0.852·22-s + 2/5·25-s + 2.35·26-s − 1.15·27-s + 0.742·29-s + 1.46·30-s − 1.41·32-s − 0.696·33-s + 2/3·36-s − 2.30·37-s + 1.94·38-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051824807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051824807\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 29 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−11.38498799057946176442410459379, −11.10968407318732394984764604351, −10.72340524029388701240238208103, −10.12135089696651518319477899204, −9.565372665576193152376368037436, −8.795662135749343224859787787849, −8.760407277076852788483618721387, −7.961999302473869907337583098222, −7.25193839178447762625809092862, −7.08489610643911451694837154361, −6.27107440988682901555371451458, −6.13540609137947951031888764138, −5.37840301739701520535674615359, −5.31875727155913726374458497847, −4.30093393857056696773949350270, −4.21219918740667359120603918234, −3.35466702775149670946345585711, −3.24224823734094687176307727001, −1.86172555398143150916783909686, −0.799977946484446644760819734528,
0.799977946484446644760819734528, 1.86172555398143150916783909686, 3.24224823734094687176307727001, 3.35466702775149670946345585711, 4.21219918740667359120603918234, 4.30093393857056696773949350270, 5.31875727155913726374458497847, 5.37840301739701520535674615359, 6.13540609137947951031888764138, 6.27107440988682901555371451458, 7.08489610643911451694837154361, 7.25193839178447762625809092862, 7.961999302473869907337583098222, 8.760407277076852788483618721387, 8.795662135749343224859787787849, 9.565372665576193152376368037436, 10.12135089696651518319477899204, 10.72340524029388701240238208103, 11.10968407318732394984764604351, 11.38498799057946176442410459379
