| L(s) = 1 | − 26·3-s + 62·5-s − 98·7-s + 198·9-s + 972·11-s − 78·13-s − 1.61e3·15-s + 560·17-s + 2.64e3·19-s + 2.54e3·21-s − 2.27e3·23-s + 1.05e3·25-s + 962·27-s + 7.80e3·29-s − 5.44e3·31-s − 2.52e4·33-s − 6.07e3·35-s − 576·37-s + 2.02e3·39-s − 1.68e4·41-s − 8.39e3·43-s + 1.22e4·45-s + 4.53e3·47-s + 7.20e3·49-s − 1.45e4·51-s − 1.42e3·53-s + 6.02e4·55-s + ⋯ |
| L(s) = 1 | − 1.66·3-s + 1.10·5-s − 0.755·7-s + 0.814·9-s + 2.42·11-s − 0.128·13-s − 1.84·15-s + 0.469·17-s + 1.67·19-s + 1.26·21-s − 0.895·23-s + 0.338·25-s + 0.253·27-s + 1.72·29-s − 1.01·31-s − 4.03·33-s − 0.838·35-s − 0.0691·37-s + 0.213·39-s − 1.56·41-s − 0.692·43-s + 0.903·45-s + 0.299·47-s + 3/7·49-s − 0.783·51-s − 0.0694·53-s + 2.68·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.357433298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357433298\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 26 T + 478 T^{2} + 26 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 62 T + 2786 T^{2} - 62 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 972 T + 551926 T^{2} - 972 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 p T - 148150 T^{2} + 6 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 560 T + 2911742 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2642 T + 6454926 T^{2} - 2642 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2272 T + 3276974 T^{2} + 2272 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7808 T + 56228822 T^{2} - 7808 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5444 T + 61676286 T^{2} + 5444 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 576 T + 63988358 T^{2} + 576 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16888 T + 277723070 T^{2} + 16888 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8396 T + 291418902 T^{2} + 8396 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4532 T + 192546782 T^{2} - 4532 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1420 T + 601997678 T^{2} + 1420 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 34146 T + 1702640302 T^{2} - 34146 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 19106 T - 202373982 T^{2} + 19106 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 56952 T + 3238977590 T^{2} - 56952 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7224 T + 3534775246 T^{2} - 7224 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 128828 T + 8014301382 T^{2} - 128828 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 52808 T + 6427138206 T^{2} + 52808 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 84486 T + 4859890798 T^{2} + 84486 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 130972 T + 14510409206 T^{2} + 130972 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 194624 T + 26622623358 T^{2} - 194624 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.35824490871653557204063131149, −10.09722694272163780500435925270, −9.642333797594491744076953674747, −9.484110496366488042071638998902, −8.807300170854804333568177762058, −8.434913414151653896298519983112, −7.58675218674520602490354732949, −6.90782291772210852074803451635, −6.59478390328425456077222121689, −6.38555066611836927786076304783, −5.67855508764826129490640752499, −5.61560310304633749081479944148, −4.99110009073321185303252887932, −4.36394503355686493807207108902, −3.43498930681517332967113222293, −3.41404759561890926887120510069, −2.20470176531966949214438510023, −1.57286552843896914422680995142, −0.931995593441399261942238545215, −0.52178033605158052394811031589,
0.52178033605158052394811031589, 0.931995593441399261942238545215, 1.57286552843896914422680995142, 2.20470176531966949214438510023, 3.41404759561890926887120510069, 3.43498930681517332967113222293, 4.36394503355686493807207108902, 4.99110009073321185303252887932, 5.61560310304633749081479944148, 5.67855508764826129490640752499, 6.38555066611836927786076304783, 6.59478390328425456077222121689, 6.90782291772210852074803451635, 7.58675218674520602490354732949, 8.434913414151653896298519983112, 8.807300170854804333568177762058, 9.484110496366488042071638998902, 9.642333797594491744076953674747, 10.09722694272163780500435925270, 10.35824490871653557204063131149
