| L(s) = 1 | + 20·5-s − 9·9-s − 60·11-s + 24·19-s + 275·25-s − 128·29-s − 624·31-s − 748·41-s − 180·45-s + 362·49-s − 1.20e3·55-s + 1.02e3·59-s − 1.50e3·61-s − 1.84e3·71-s − 144·79-s + 81·81-s + 580·89-s + 480·95-s + 540·99-s − 208·101-s + 1.14e3·109-s + 38·121-s + 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 1.64·11-s + 0.289·19-s + 11/5·25-s − 0.819·29-s − 3.61·31-s − 2.84·41-s − 0.596·45-s + 1.05·49-s − 2.94·55-s + 2.25·59-s − 3.16·61-s − 3.08·71-s − 0.205·79-s + 1/9·81-s + 0.690·89-s + 0.518·95-s + 0.548·99-s − 0.204·101-s + 1.00·109-s + 0.0285·121-s + 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5443540000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5443540000\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 362 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4926 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 82262 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 374 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 60010 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 190222 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 98838 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 754 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 454070 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 924 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 662434 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1119238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1683970 T^{2} + p^{6} T^{4} \) |
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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
−9.883747043869264609616803354661, −9.437414796524879927281814537300, −8.966807818755541264638474598656, −8.825724197138369662109804312543, −8.319365637100553581180376569481, −7.49394299221225721600407253755, −7.43339206103494082802631976089, −6.97499238785242394662478021938, −6.28493853677735342957619913561, −5.79138138785060594419899892078, −5.60933703514232340206117648676, −5.05469495336752981402352651433, −5.01443186540201151085891352079, −3.99414794664221828007625843856, −3.36037376127796979230892525851, −2.93147558592401525507769627392, −2.29735077190688654316820114469, −1.81730297953678121056288010988, −1.43869291161114602974412214960, −0.16964588424100884333697373705,
0.16964588424100884333697373705, 1.43869291161114602974412214960, 1.81730297953678121056288010988, 2.29735077190688654316820114469, 2.93147558592401525507769627392, 3.36037376127796979230892525851, 3.99414794664221828007625843856, 5.01443186540201151085891352079, 5.05469495336752981402352651433, 5.60933703514232340206117648676, 5.79138138785060594419899892078, 6.28493853677735342957619913561, 6.97499238785242394662478021938, 7.43339206103494082802631976089, 7.49394299221225721600407253755, 8.319365637100553581180376569481, 8.825724197138369662109804312543, 8.966807818755541264638474598656, 9.437414796524879927281814537300, 9.883747043869264609616803354661
