| L(s) = 1 | − 2·3-s − 4-s − 5-s + 3·9-s + 11-s + 2·12-s + 2·15-s + 16-s + 20-s − 7·23-s − 3·25-s − 4·27-s + 31-s − 2·33-s − 3·36-s + 7·37-s − 44-s − 3·45-s + 13·47-s − 2·48-s + 49-s − 5·53-s − 55-s + 3·59-s − 2·60-s − 64-s + 12·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.447·5-s + 9-s + 0.301·11-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 0.223·20-s − 1.45·23-s − 3/5·25-s − 0.769·27-s + 0.179·31-s − 0.348·33-s − 1/2·36-s + 1.15·37-s − 0.150·44-s − 0.447·45-s + 1.89·47-s − 0.288·48-s + 1/7·49-s − 0.686·53-s − 0.134·55-s + 0.390·59-s − 0.258·60-s − 1/8·64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−7.52762712697587657263010644471, −7.00402164374025168755318548697, −6.54214085601683384907620106545, −6.10878976323756887844324767104, −5.77503041166670714267004163067, −5.43400243199392372637399321908, −4.87572181271043075163269651748, −4.28269410667252079460638359096, −4.13789819548286247267062318976, −3.71850034705211522774469739571, −2.93406378582761935887723097576, −2.27134608949972785456222357257, −1.54393518050178245150443581761, −0.791960728109073280055411640713, 0,
0.791960728109073280055411640713, 1.54393518050178245150443581761, 2.27134608949972785456222357257, 2.93406378582761935887723097576, 3.71850034705211522774469739571, 4.13789819548286247267062318976, 4.28269410667252079460638359096, 4.87572181271043075163269651748, 5.43400243199392372637399321908, 5.77503041166670714267004163067, 6.10878976323756887844324767104, 6.54214085601683384907620106545, 7.00402164374025168755318548697, 7.52762712697587657263010644471
