| L(s) = 1 | − 8·3-s + 10·5-s − 8·7-s + 18·9-s − 64·11-s + 12·13-s − 80·15-s + 4·17-s − 208·19-s + 64·21-s − 120·23-s + 75·25-s + 8·27-s − 292·29-s − 176·31-s + 512·33-s − 80·35-s − 356·37-s − 96·39-s + 100·41-s + 376·43-s + 180·45-s + 280·47-s − 38·49-s − 32·51-s + 316·53-s − 640·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.894·5-s − 0.431·7-s + 2/3·9-s − 1.75·11-s + 0.256·13-s − 1.37·15-s + 0.0570·17-s − 2.51·19-s + 0.665·21-s − 1.08·23-s + 3/5·25-s + 0.0570·27-s − 1.86·29-s − 1.01·31-s + 2.70·33-s − 0.386·35-s − 1.58·37-s − 0.394·39-s + 0.380·41-s + 1.33·43-s + 0.596·45-s + 0.868·47-s − 0.110·49-s − 0.0878·51-s + 0.818·53-s − 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 64 T + 2822 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 T + 2894 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T - 3994 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 208 T + 22998 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 120 T + 25030 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 292 T + 63950 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 176 T + 66462 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 356 T + 126846 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 100 T + 101942 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161502 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 280 T + 223190 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 316 T + 308894 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 720 T + 530758 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 744 T + 686894 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 48 T - 424178 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 940 T + 653334 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32 T + 276318 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1592 T + 1724174 T^{2} - 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 780 T + 1506742 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1220 T + 2159046 T^{2} - 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.25947462320336686498013024916, −11.67288393937301171034241677594, −10.77572968474054492585080681296, −10.71791211582910273354148816421, −10.59557007474207693286623356113, −9.831397527990068891311537236633, −9.015053537709945591784015547806, −8.825722889401374721802754932065, −7.72909516862182952904923820748, −7.51733194568139722651819832651, −6.37498946771506947688090822251, −6.24557936607220958529091945617, −5.57499802842084390258093076947, −5.36474528289439355992182085210, −4.49364377648893634558864690912, −3.67763642061837151954802063857, −2.46770584363427504292004799203, −1.86515967599364179380502323219, 0, 0,
1.86515967599364179380502323219, 2.46770584363427504292004799203, 3.67763642061837151954802063857, 4.49364377648893634558864690912, 5.36474528289439355992182085210, 5.57499802842084390258093076947, 6.24557936607220958529091945617, 6.37498946771506947688090822251, 7.51733194568139722651819832651, 7.72909516862182952904923820748, 8.825722889401374721802754932065, 9.015053537709945591784015547806, 9.831397527990068891311537236633, 10.59557007474207693286623356113, 10.71791211582910273354148816421, 10.77572968474054492585080681296, 11.67288393937301171034241677594, 12.25947462320336686498013024916
