| L(s) = 1 | − 2·2-s − 2·4-s + 8·8-s + 32·11-s − 4·16-s + 8·17-s + 32·19-s − 64·22-s + 28·25-s + 8·32-s − 16·34-s − 64·38-s − 40·41-s + 32·43-s − 64·44-s + 76·49-s − 56·50-s + 128·59-s − 24·64-s − 256·67-s − 16·68-s + 200·73-s − 64·76-s + 80·82-s − 160·83-s − 64·86-s + 256·88-s + ⋯ |
| L(s) = 1 | − 2-s − 1/2·4-s + 8-s + 2.90·11-s − 1/4·16-s + 8/17·17-s + 1.68·19-s − 2.90·22-s + 1.11·25-s + 1/4·32-s − 0.470·34-s − 1.68·38-s − 0.975·41-s + 0.744·43-s − 1.45·44-s + 1.55·49-s − 1.11·50-s + 2.16·59-s − 3/8·64-s − 3.82·67-s − 0.235·68-s + 2.73·73-s − 0.842·76-s + 0.975·82-s − 1.92·83-s − 0.744·86-s + 2.90·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.105789756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105789756\) |
| \(L(2)\) |
|
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 | $D_{4}$ | \( 1 + p T + 3 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 390 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 738 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 20 T + 1734 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 64 T + 7554 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 80 T + 14610 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 100 T + 11430 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−10.66160014150586465012153707820, −10.06810863887070178928168700290, −9.971603782805417757687528382555, −9.706635162918568636934257547950, −9.318578485766785567438175810599, −9.080282463822533586893818136635, −8.835664394686978607419505281794, −8.622317305641944247219840581777, −8.542447262057185222569729469236, −7.71314935488871183209196092204, −7.49800902505051585275999164462, −7.39061357997135212246354669911, −6.83931481435058043642446244422, −6.48455552306572344912327671009, −6.24629454738999013042835482030, −5.77324905175859997564278806102, −5.31060905666380159580658421642, −4.81769761580209726993722822834, −4.57227852994403919594246230971, −3.86097316195256239732338945321, −3.72827860066597379228394283970, −3.23045521285636641816806455755, −2.36620447646403876290220758775, −1.22176471274579428101941551224, −1.03992153995220372661567779604,
1.03992153995220372661567779604, 1.22176471274579428101941551224, 2.36620447646403876290220758775, 3.23045521285636641816806455755, 3.72827860066597379228394283970, 3.86097316195256239732338945321, 4.57227852994403919594246230971, 4.81769761580209726993722822834, 5.31060905666380159580658421642, 5.77324905175859997564278806102, 6.24629454738999013042835482030, 6.48455552306572344912327671009, 6.83931481435058043642446244422, 7.39061357997135212246354669911, 7.49800902505051585275999164462, 7.71314935488871183209196092204, 8.542447262057185222569729469236, 8.622317305641944247219840581777, 8.835664394686978607419505281794, 9.080282463822533586893818136635, 9.318578485766785567438175810599, 9.706635162918568636934257547950, 9.971603782805417757687528382555, 10.06810863887070178928168700290, 10.66160014150586465012153707820
