| L(s) = 1 | − 144·11-s − 104·19-s − 156·29-s + 240·31-s − 724·41-s + 670·49-s + 1.39e3·59-s + 444·61-s − 192·71-s + 1.26e3·79-s + 1.98e3·89-s − 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.94·11-s − 1.25·19-s − 0.998·29-s + 1.39·31-s − 2.75·41-s + 1.95·49-s + 3.07·59-s + 0.931·61-s − 0.320·71-s + 1.80·79-s + 2.36·89-s − 1.75·101-s − 0.783·109-s + 9.68·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.504305627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504305627\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1230 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 78806 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 362 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 75242 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 129246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 151146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 696 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 601510 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 746350 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 994 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 844610 T^{2} + p^{6} T^{4} \) |
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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
−8.769624283951839779224161818360, −8.729506132722121740935377533976, −8.252461804679939124160849011687, −8.021432699691598130693618774215, −7.47366694378677567527593638023, −7.41846790654937294001570853889, −6.61374709425533344248739659561, −6.47945172842430259768838614561, −5.67865959901195474963839717366, −5.38820423823188472610113684890, −5.04527079815134221666517897767, −4.93764322837246195963958473084, −4.04455543653042023952337869989, −3.77153685162631980518425940141, −2.86872127922451614923331089956, −2.77872204378069124863887952772, −2.12657878612317860353125078668, −1.95035263958724987888613994055, −0.60467369287496069402479059828, −0.40935333957269288356934686551,
0.40935333957269288356934686551, 0.60467369287496069402479059828, 1.95035263958724987888613994055, 2.12657878612317860353125078668, 2.77872204378069124863887952772, 2.86872127922451614923331089956, 3.77153685162631980518425940141, 4.04455543653042023952337869989, 4.93764322837246195963958473084, 5.04527079815134221666517897767, 5.38820423823188472610113684890, 5.67865959901195474963839717366, 6.47945172842430259768838614561, 6.61374709425533344248739659561, 7.41846790654937294001570853889, 7.47366694378677567527593638023, 8.021432699691598130693618774215, 8.252461804679939124160849011687, 8.729506132722121740935377533976, 8.769624283951839779224161818360
