| L(s) = 1 | + 3·5-s + 9·7-s + 80·11-s − 26·13-s − 19·17-s + 84·19-s + 196·23-s − 239·25-s + 44·29-s + 86·31-s + 27·35-s + 209·37-s + 230·41-s − 287·43-s + 435·47-s − 111·49-s + 118·53-s + 240·55-s − 368·59-s − 1.05e3·61-s − 78·65-s − 68·67-s − 131·71-s + 456·73-s + 720·77-s + 1.00e3·79-s + 1.95e3·83-s + ⋯ |
| L(s) = 1 | + 0.268·5-s + 0.485·7-s + 2.19·11-s − 0.554·13-s − 0.271·17-s + 1.01·19-s + 1.77·23-s − 1.91·25-s + 0.281·29-s + 0.498·31-s + 0.130·35-s + 0.928·37-s + 0.876·41-s − 1.01·43-s + 1.35·47-s − 0.323·49-s + 0.305·53-s + 0.588·55-s − 0.812·59-s − 2.22·61-s − 0.148·65-s − 0.123·67-s − 0.218·71-s + 0.731·73-s + 1.06·77-s + 1.43·79-s + 2.58·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.318260859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.318260859\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 192 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19 T + 8688 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 56518 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 230 T + 149010 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 287 T + 92698 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1008 T + 1233294 T^{2} - 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 720 T + 899726 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.268046566842532199628261968282, −8.877667378776072116399103378004, −8.095686625600985190607238115265, −8.027792599297560308803548669388, −7.39119480479793905507128861275, −7.20808658495018792185795251626, −6.49699834950225082335173118386, −6.49471910399457326847249914479, −5.82717405675198499728833220527, −5.59376953278484191834531074613, −4.75815501920695719493700443310, −4.73095676189754720910215757115, −4.14463722450932306991875370827, −3.68274495874402225753070697839, −3.18254610987168635511646159571, −2.72717807223410665638655251605, −1.83814564341314226195667954201, −1.75275396350090846567808787996, −0.853712218311436120569411009554, −0.68878274110272297712467633036,
0.68878274110272297712467633036, 0.853712218311436120569411009554, 1.75275396350090846567808787996, 1.83814564341314226195667954201, 2.72717807223410665638655251605, 3.18254610987168635511646159571, 3.68274495874402225753070697839, 4.14463722450932306991875370827, 4.73095676189754720910215757115, 4.75815501920695719493700443310, 5.59376953278484191834531074613, 5.82717405675198499728833220527, 6.49471910399457326847249914479, 6.49699834950225082335173118386, 7.20808658495018792185795251626, 7.39119480479793905507128861275, 8.027792599297560308803548669388, 8.095686625600985190607238115265, 8.877667378776072116399103378004, 9.268046566842532199628261968282
