| L(s) = 1 | − 28·9-s + 280·17-s + 140·25-s + 728·41-s − 348·49-s − 3.64e3·73-s − 870·81-s + 2.18e3·89-s − 1.96e3·97-s − 3.64e3·113-s − 1.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 7.84e3·153-s + 157-s + 163-s + 167-s − 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.03·9-s + 3.99·17-s + 1.11·25-s + 2.77·41-s − 1.01·49-s − 5.83·73-s − 1.19·81-s + 2.60·89-s − 2.05·97-s − 3.03·113-s − 1.05·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 4.14·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.855981221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855981221\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 14 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 174 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 702 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 4074 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 6958 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 754 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 33098 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 39814 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 141374 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 107294 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 282074 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 403998 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 399882 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 552526 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 703022 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 910 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 525278 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 632814 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 546 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.448783195659335254461678674905, −7.77110021882884065442369727389, −7.74066911150519706963832341416, −7.54401535860393283189289123736, −7.52338215833971042927752962510, −7.12573394853413171373022686224, −6.51686547269716399548745856877, −6.39289495823577702074046085854, −6.12383254673801080859907733760, −5.75538755066282345655153834152, −5.51478642967468025521835548854, −5.46990303725740127031856136651, −5.06133754337830436455469454741, −4.78183953097813453930788908699, −4.27179668937744021307156882301, −3.94581939992963004162318102018, −3.76722166643246114933524129058, −3.06148260417604879894659198697, −3.05222737701110933557038753359, −2.80810883829530013745395077812, −2.45126515559611105134588950925, −1.51754484392689165697370344442, −1.22451930226900963150378655118, −1.08536858699280234727273073805, −0.25460237163787471317467428679,
0.25460237163787471317467428679, 1.08536858699280234727273073805, 1.22451930226900963150378655118, 1.51754484392689165697370344442, 2.45126515559611105134588950925, 2.80810883829530013745395077812, 3.05222737701110933557038753359, 3.06148260417604879894659198697, 3.76722166643246114933524129058, 3.94581939992963004162318102018, 4.27179668937744021307156882301, 4.78183953097813453930788908699, 5.06133754337830436455469454741, 5.46990303725740127031856136651, 5.51478642967468025521835548854, 5.75538755066282345655153834152, 6.12383254673801080859907733760, 6.39289495823577702074046085854, 6.51686547269716399548745856877, 7.12573394853413171373022686224, 7.52338215833971042927752962510, 7.54401535860393283189289123736, 7.74066911150519706963832341416, 7.77110021882884065442369727389, 8.448783195659335254461678674905
