| L(s) = 1 | − 2·2-s − 4·3-s − 4·5-s + 8·6-s − 3·7-s + 4·8-s + 6·9-s + 8·10-s − 6·11-s + 6·14-s + 16·15-s − 4·16-s − 4·17-s − 12·18-s − 12·19-s + 12·21-s + 12·22-s − 2·23-s − 16·24-s + 3·25-s + 4·27-s − 8·29-s − 32·30-s − 16·31-s + 24·33-s + 8·34-s + 12·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 1.78·5-s + 3.26·6-s − 1.13·7-s + 1.41·8-s + 2·9-s + 2.52·10-s − 1.80·11-s + 1.60·14-s + 4.13·15-s − 16-s − 0.970·17-s − 2.82·18-s − 2.75·19-s + 2.61·21-s + 2.55·22-s − 0.417·23-s − 3.26·24-s + 3/5·25-s + 0.769·27-s − 1.48·29-s − 5.84·30-s − 2.87·31-s + 4.17·33-s + 1.37·34-s + 2.02·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71407 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71407 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 101 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−15.5073333284, −14.8586787590, −14.3011595500, −13.2171643863, −13.0050821702, −12.6928483729, −12.4638189837, −11.5771973615, −11.1898795365, −11.0578070081, −10.6403180689, −10.2420217692, −9.63988657265, −9.06031789874, −8.54824564197, −8.02712745417, −7.74187934910, −7.07842349991, −6.34404657279, −6.16055116458, −5.17330498735, −4.98916157968, −4.07610506844, −3.72933468340, −2.28603932081, 0, 0, 0,
2.28603932081, 3.72933468340, 4.07610506844, 4.98916157968, 5.17330498735, 6.16055116458, 6.34404657279, 7.07842349991, 7.74187934910, 8.02712745417, 8.54824564197, 9.06031789874, 9.63988657265, 10.2420217692, 10.6403180689, 11.0578070081, 11.1898795365, 11.5771973615, 12.4638189837, 12.6928483729, 13.0050821702, 13.2171643863, 14.3011595500, 14.8586787590, 15.5073333284
