| L(s) = 1 | − 2·2-s − 10·4-s − 10·5-s + 32·8-s + 20·10-s + 22·11-s − 64·13-s + 44·16-s − 32·17-s − 10·19-s + 100·20-s − 44·22-s + 84·23-s + 75·25-s + 128·26-s + 170·29-s − 258·31-s − 248·32-s + 64·34-s + 76·37-s + 20·38-s − 320·40-s + 578·41-s + 380·43-s − 220·44-s − 168·46-s + 484·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 5/4·4-s − 0.894·5-s + 1.41·8-s + 0.632·10-s + 0.603·11-s − 1.36·13-s + 0.687·16-s − 0.456·17-s − 0.120·19-s + 1.11·20-s − 0.426·22-s + 0.761·23-s + 3/5·25-s + 0.965·26-s + 1.08·29-s − 1.49·31-s − 1.37·32-s + 0.322·34-s + 0.337·37-s + 0.0853·38-s − 1.26·40-s + 2.20·41-s + 1.34·43-s − 0.753·44-s − 0.538·46-s + 1.50·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9844207783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9844207783\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 7 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 638 T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 431 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 64 T + 5370 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 32 T + 674 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T - 129 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 84 T + 23026 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 170 T + 54275 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 258 T + 75791 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 76 T + 37038 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 578 T + 214451 T^{2} - 578 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 380 T + 122106 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 484 T + 265442 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 544 T + 341738 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 706 T + 533015 T^{2} - 706 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 668 T + 454926 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1452 T + 1120490 T^{2} + 1452 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 974 T + 837743 T^{2} - 974 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1184 T + 1103106 T^{2} - 1184 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 995246 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 444 T + 112858 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 513 T + p^{3} T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 668 T + 1356102 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.89924705050988028170888096962, −10.71219876121537935078554994302, −9.841003726026216146639662146285, −9.694911450770949298246618180317, −9.139696903754866133881869430916, −8.903995345229456521852935828896, −8.399838818762993355120355919752, −7.925054545197753426535918715729, −7.35230830142601247188023136304, −7.20585487983406130434073533120, −6.45829167720504134291313476563, −5.69146875848402258797863175685, −5.06898023368518580619222742805, −4.67926562738334554723928375834, −4.06809010344881231755725348998, −3.83652209701108997212525189487, −2.83311436785666374851377459104, −2.10680169861293815281563929351, −0.71682207030758060382717587226, −0.65566503017852838083783047216,
0.65566503017852838083783047216, 0.71682207030758060382717587226, 2.10680169861293815281563929351, 2.83311436785666374851377459104, 3.83652209701108997212525189487, 4.06809010344881231755725348998, 4.67926562738334554723928375834, 5.06898023368518580619222742805, 5.69146875848402258797863175685, 6.45829167720504134291313476563, 7.20585487983406130434073533120, 7.35230830142601247188023136304, 7.925054545197753426535918715729, 8.399838818762993355120355919752, 8.903995345229456521852935828896, 9.139696903754866133881869430916, 9.694911450770949298246618180317, 9.841003726026216146639662146285, 10.71219876121537935078554994302, 10.89924705050988028170888096962
