| L(s) = 1 | − 3·3-s + 3·4-s + 5·5-s + 4·9-s − 11-s − 9·12-s − 15·15-s + 5·16-s + 15·20-s − 6·23-s + 16·25-s − 4·31-s + 3·33-s + 12·36-s − 10·37-s − 3·44-s + 20·45-s − 8·47-s − 15·48-s − 10·53-s − 5·55-s − 6·59-s − 45·60-s + 3·64-s + 2·67-s + 18·69-s + 14·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 3/2·4-s + 2.23·5-s + 4/3·9-s − 0.301·11-s − 2.59·12-s − 3.87·15-s + 5/4·16-s + 3.35·20-s − 1.25·23-s + 16/5·25-s − 0.718·31-s + 0.522·33-s + 2·36-s − 1.64·37-s − 0.452·44-s + 2.98·45-s − 1.16·47-s − 2.16·48-s − 1.37·53-s − 0.674·55-s − 0.781·59-s − 5.80·60-s + 3/8·64-s + 0.244·67-s + 2.16·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19965 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19965 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.268514887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268514887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.80712580042631435895918114826, −10.51030870003857717170792171854, −10.05395702900173771309262258302, −9.588876269270554127406339657975, −8.867844072189662011395620793102, −7.976391008460973429246898512500, −7.17432578348212283038458097542, −6.53320134550634356808509964251, −6.32092094732012211563375513099, −5.85747864181885621056541117646, −5.31448672059726801298323261474, −4.85634057575082581314782853033, −3.27548926464812173941221289856, −2.20091566131228422932655094636, −1.64189346126065853763241847449,
1.64189346126065853763241847449, 2.20091566131228422932655094636, 3.27548926464812173941221289856, 4.85634057575082581314782853033, 5.31448672059726801298323261474, 5.85747864181885621056541117646, 6.32092094732012211563375513099, 6.53320134550634356808509964251, 7.17432578348212283038458097542, 7.976391008460973429246898512500, 8.867844072189662011395620793102, 9.588876269270554127406339657975, 10.05395702900173771309262258302, 10.51030870003857717170792171854, 10.80712580042631435895918114826
