| L(s) = 1 | + 16·5-s − 10·9-s − 8·13-s − 4·17-s + 142·25-s + 28·29-s + 28·37-s + 92·41-s − 160·45-s − 7·49-s − 44·53-s + 96·61-s − 128·65-s − 220·73-s + 19·81-s − 64·85-s − 268·89-s − 356·97-s + 80·101-s + 364·109-s − 116·113-s + 80·117-s + 130·121-s + 848·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 16/5·5-s − 1.11·9-s − 0.615·13-s − 0.235·17-s + 5.67·25-s + 0.965·29-s + 0.756·37-s + 2.24·41-s − 3.55·45-s − 1/7·49-s − 0.830·53-s + 1.57·61-s − 1.96·65-s − 3.01·73-s + 0.234·81-s − 0.752·85-s − 3.01·89-s − 3.67·97-s + 0.792·101-s + 3.33·109-s − 1.02·113-s + 0.683·117-s + 1.07·121-s + 6.78·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.612416344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.612416344\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 130 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 610 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 914 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3586 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3410 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )( 1 + 118 T + p^{2} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 114 T + p^{2} T^{2} )( 1 + 114 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 12406 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 134 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{2} T^{2} )^{2} \) |
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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
−13.94685507212963549163486608800, −13.21432697895065728119013912910, −12.73452966790673885555518562537, −12.38001279636780710645655746634, −11.28358800586661379073193097480, −11.09040867852138910565025180795, −10.21336811093121172273189282744, −9.818856830119417266232557690101, −9.654019853187693809039692158927, −8.801801340347135863674983002998, −8.645744602984799093649647491056, −7.52169407207388329708972029313, −6.70667778550050906414932982390, −6.09186956031030647764937222976, −5.78857047819066157170620777864, −5.29344526219146188010973232090, −4.46755723642027647351788651704, −2.64391421228244465049837202373, −2.61737207192282527323600793105, −1.42992247741257885403348369265,
1.42992247741257885403348369265, 2.61737207192282527323600793105, 2.64391421228244465049837202373, 4.46755723642027647351788651704, 5.29344526219146188010973232090, 5.78857047819066157170620777864, 6.09186956031030647764937222976, 6.70667778550050906414932982390, 7.52169407207388329708972029313, 8.645744602984799093649647491056, 8.801801340347135863674983002998, 9.654019853187693809039692158927, 9.818856830119417266232557690101, 10.21336811093121172273189282744, 11.09040867852138910565025180795, 11.28358800586661379073193097480, 12.38001279636780710645655746634, 12.73452966790673885555518562537, 13.21432697895065728119013912910, 13.94685507212963549163486608800
