| L(s) = 1 | − 10·5-s − 15·7-s + 45·11-s − 26·13-s + 5·17-s − 92·19-s + 173·23-s + 75·25-s + 90·29-s − 242·31-s + 150·35-s − 155·37-s − 9·41-s − 384·43-s + 186·47-s − 415·49-s − 209·53-s − 450·55-s + 596·59-s + 243·61-s + 260·65-s − 500·67-s − 307·71-s + 106·73-s − 675·77-s + 81·79-s + 720·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.809·7-s + 1.23·11-s − 0.554·13-s + 0.0713·17-s − 1.11·19-s + 1.56·23-s + 3/5·25-s + 0.576·29-s − 1.40·31-s + 0.724·35-s − 0.688·37-s − 0.0342·41-s − 1.36·43-s + 0.577·47-s − 1.20·49-s − 0.541·53-s − 1.10·55-s + 1.31·59-s + 0.510·61-s + 0.496·65-s − 0.911·67-s − 0.513·71-s + 0.169·73-s − 0.999·77-s + 0.115·79-s + 0.952·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.597708257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597708257\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 15 T + 640 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 45 T + 3066 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 1550 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 92 T + 9290 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 173 T + 632 p T^{2} - 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 90 T + 50394 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 242 T + 54182 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 155 T + 94940 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 108312 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 384 T + 189334 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 186 T + 124270 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 209 T + 279124 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 596 T + 497926 T^{2} - 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 243 T + 466168 T^{2} - 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 500 T + 466070 T^{2} + 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 307 T + 567502 T^{2} + 307 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 106 T + 662642 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 81 T + 970438 T^{2} - 81 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 720 T + 1214278 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 613 T + 1502960 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 387 T + 1808698 T^{2} + 387 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.843436844751078106229125657356, −8.590362743569335544220458860683, −7.899916738307647563041822948668, −7.85212913460084483771073919665, −7.05223081637029817144724657373, −6.96211601575023458410654801605, −6.50134734325900855098207399994, −6.45558491597455237698212380933, −5.60782703295480722494672173602, −5.29696320660266747957712072771, −4.79187766266362926516790921787, −4.39015820394975576621304864893, −3.91646405924824025701256967826, −3.58513350453044507971724757122, −3.07339462464699789636570611598, −2.81447906117430639148266199857, −1.89622316436968449872689247230, −1.61020601335783307891243995298, −0.73755781874062339368655728491, −0.34404175462039614338697922095,
0.34404175462039614338697922095, 0.73755781874062339368655728491, 1.61020601335783307891243995298, 1.89622316436968449872689247230, 2.81447906117430639148266199857, 3.07339462464699789636570611598, 3.58513350453044507971724757122, 3.91646405924824025701256967826, 4.39015820394975576621304864893, 4.79187766266362926516790921787, 5.29696320660266747957712072771, 5.60782703295480722494672173602, 6.45558491597455237698212380933, 6.50134734325900855098207399994, 6.96211601575023458410654801605, 7.05223081637029817144724657373, 7.85212913460084483771073919665, 7.899916738307647563041822948668, 8.590362743569335544220458860683, 8.843436844751078106229125657356
