| L(s) = 1 | − 4·4-s + 4·5-s − 9·9-s + 140·11-s + 16·16-s − 48·19-s − 16·20-s − 109·25-s − 432·29-s + 416·31-s + 36·36-s − 412·41-s − 560·44-s − 36·45-s + 682·49-s + 560·55-s + 740·59-s − 1.10e3·61-s − 64·64-s − 1.08e3·71-s + 192·76-s − 1.58e3·79-s + 64·80-s + 81·81-s + 1.87e3·89-s − 192·95-s − 1.26e3·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.357·5-s − 1/3·9-s + 3.83·11-s + 1/4·16-s − 0.579·19-s − 0.178·20-s − 0.871·25-s − 2.76·29-s + 2.41·31-s + 1/6·36-s − 1.56·41-s − 1.91·44-s − 0.119·45-s + 1.98·49-s + 1.37·55-s + 1.63·59-s − 2.30·61-s − 1/8·64-s − 1.80·71-s + 0.289·76-s − 2.25·79-s + 0.0894·80-s + 1/9·81-s + 2.23·89-s − 0.207·95-s − 1.27·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.435441197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435441197\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9342 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14334 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 216 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 208 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 206 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 105246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 370 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 550 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 71542 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 540 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 413218 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 980358 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1822210 T^{2} + p^{6} T^{4} \) |
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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
−16.92596340143600097357538620043, −16.66459437335727761570599754612, −15.42885686336603796306432432538, −14.69583482760204588594300735012, −14.60016700086929928390528059734, −13.64725299190684495126542792807, −13.51502361570863337053377513253, −12.28638532742863199342194054136, −11.72244892902209453657110899494, −11.53388052525637704296874415528, −10.29804138956363154700175073752, −9.511412258802902153774109227034, −9.059799341177537628681199805640, −8.563355162756076935360265007437, −7.25420210453856398975202879064, −6.41556334946495839028693815823, −5.86132693280654407960530228253, −4.30556151192794662211149799007, −3.73327048352460643421953506453, −1.51793441717762989104647157434,
1.51793441717762989104647157434, 3.73327048352460643421953506453, 4.30556151192794662211149799007, 5.86132693280654407960530228253, 6.41556334946495839028693815823, 7.25420210453856398975202879064, 8.563355162756076935360265007437, 9.059799341177537628681199805640, 9.511412258802902153774109227034, 10.29804138956363154700175073752, 11.53388052525637704296874415528, 11.72244892902209453657110899494, 12.28638532742863199342194054136, 13.51502361570863337053377513253, 13.64725299190684495126542792807, 14.60016700086929928390528059734, 14.69583482760204588594300735012, 15.42885686336603796306432432538, 16.66459437335727761570599754612, 16.92596340143600097357538620043
