| L(s) = 1 | + 40·3-s + 280·7-s − 653·9-s + 2.28e3·11-s − 5.36e3·13-s + 2.91e4·17-s + 2.09e4·19-s + 1.12e4·21-s + 1.10e4·23-s − 2.87e4·27-s − 2.30e5·29-s + 2.14e5·31-s + 9.12e4·33-s + 7.34e5·37-s − 2.14e5·39-s + 4.49e5·41-s + 8.31e5·43-s + 1.97e6·47-s − 1.49e6·49-s + 1.16e6·51-s + 4.56e5·53-s + 8.38e5·57-s − 9.43e5·59-s + 1.26e6·61-s − 1.82e5·63-s + 3.14e6·67-s + 4.41e5·69-s + ⋯ |
| L(s) = 1 | + 0.855·3-s + 0.308·7-s − 0.298·9-s + 0.516·11-s − 0.676·13-s + 1.43·17-s + 0.701·19-s + 0.263·21-s + 0.189·23-s − 0.281·27-s − 1.75·29-s + 1.29·31-s + 0.441·33-s + 2.38·37-s − 0.578·39-s + 1.01·41-s + 1.59·43-s + 2.77·47-s − 1.81·49-s + 1.22·51-s + 0.421·53-s + 0.599·57-s − 0.598·59-s + 0.714·61-s − 0.0921·63-s + 1.27·67-s + 0.161·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.644104394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.644104394\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 40 T + 751 p T^{2} - 40 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 40 p T + 1575930 T^{2} - 40 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2280 T + 35168917 T^{2} - 2280 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5360 T + 71328378 T^{2} + 5360 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 29130 T + 562337467 T^{2} - 29130 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 20968 T - 76852491 T^{2} - 20968 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 480 p T + 6479910730 T^{2} - 480 p^{8} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 230952 T + 37951130794 T^{2} + 230952 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 214672 T + 66489522618 T^{2} - 214672 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 734620 T + 309225174990 T^{2} - 734620 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 449082 T + 267091482043 T^{2} - 449082 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 831520 T + 596468789814 T^{2} - 831520 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1975800 T + 1935662583982 T^{2} - 1975800 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 456780 T + 1054955456350 T^{2} - 456780 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 943824 T + 1101608154982 T^{2} + 943824 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1266364 T + 5241560521566 T^{2} - 1266364 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3142120 T + 14452012888005 T^{2} - 3142120 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3616776 T + 18754925905726 T^{2} - 3616776 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 640390 T + 19515473363643 T^{2} - 640390 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3039896 T + 32690326402122 T^{2} + 3039896 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 395400 T + 39547248930445 T^{2} + 395400 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3239262 T + 52422947109619 T^{2} - 3239262 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12842540 T + 192014878082790 T^{2} + 12842540 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.54526407941656335210571062840, −12.44696034874323735750255692141, −11.58647467796534896386652475882, −11.28280221266058219235804681791, −10.62783602520039192195396901314, −9.817754400202378065886826645607, −9.348971488222383494846716027368, −9.192656034781671515924633948499, −8.103095451752771005419103778345, −7.909730934913044761165444949140, −7.39933588277481883139391087310, −6.57722986929665522852222596643, −5.68181595979646761686470795161, −5.38682268280386057629068175331, −4.26986311058077895323179629837, −3.81843972938883060743378113562, −2.70709838982182714032734371616, −2.59513738061935786586777584722, −1.29987749380306401416438308106, −0.69694962984193693070597878170,
0.69694962984193693070597878170, 1.29987749380306401416438308106, 2.59513738061935786586777584722, 2.70709838982182714032734371616, 3.81843972938883060743378113562, 4.26986311058077895323179629837, 5.38682268280386057629068175331, 5.68181595979646761686470795161, 6.57722986929665522852222596643, 7.39933588277481883139391087310, 7.909730934913044761165444949140, 8.103095451752771005419103778345, 9.192656034781671515924633948499, 9.348971488222383494846716027368, 9.817754400202378065886826645607, 10.62783602520039192195396901314, 11.28280221266058219235804681791, 11.58647467796534896386652475882, 12.44696034874323735750255692141, 12.54526407941656335210571062840
