L(s) = 1 | − 6·3-s + 14·7-s + 27·9-s + 54·11-s + 74·13-s + 98·17-s + 42·19-s − 84·21-s + 58·23-s − 108·27-s + 248·29-s + 216·31-s − 324·33-s + 242·37-s − 444·39-s − 192·41-s + 24·43-s + 582·47-s + 147·49-s − 588·51-s − 112·53-s − 252·57-s − 746·59-s + 52·61-s + 378·63-s + 684·67-s − 348·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.755·7-s + 9-s + 1.48·11-s + 1.57·13-s + 1.39·17-s + 0.507·19-s − 0.872·21-s + 0.525·23-s − 0.769·27-s + 1.58·29-s + 1.25·31-s − 1.70·33-s + 1.07·37-s − 1.82·39-s − 0.731·41-s + 0.0851·43-s + 1.80·47-s + 3/7·49-s − 1.61·51-s − 0.290·53-s − 0.585·57-s − 1.64·59-s + 0.109·61-s + 0.755·63-s + 1.24·67-s − 0.607·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.112166262\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.112166262\) |
\(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
good | 11 | $D_{4}$ | \( 1 - 54 T + 2955 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 74 T + 3038 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 98 T + 9502 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 42 T + 5330 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 58 T - 2729 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 248 T + 61429 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 216 T + 60346 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 242 T + 80631 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 192 T + 140082 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 24 T + 9937 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 582 T + 279138 T^{2} - 582 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 112 T + 265574 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 746 T + 470426 T^{2} + 746 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 52 T + 433274 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 684 T + 717509 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1010 T + 959947 T^{2} + 1010 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 758 T + 903254 T^{2} + 758 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1420 T + 1465653 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 616 T + 104402 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 390 T + 1434774 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 824 T + 1520286 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.839020637679438962438054246907, −8.660681346588510715851278661209, −8.028316523365771278888743096194, −7.922065897600999570356155061410, −7.24470093947686550662384877534, −6.97584779158917003164441571871, −6.43505295599236003113349769683, −6.21583090682654155982218549237, −5.70397031355003125619207722594, −5.63016141333174865729898076790, −4.75833103936539815706254760731, −4.69782393429834785439897408207, −4.02353898938027299224793155404, −3.86616723306911510718976498766, −3.03231072742430237417772370014, −2.82799271880609039780699697083, −1.69122191498879502439442344876, −1.32552411364965075653767605229, −1.00949073374241936151229301270, −0.63527393943777789908092387398,
0.63527393943777789908092387398, 1.00949073374241936151229301270, 1.32552411364965075653767605229, 1.69122191498879502439442344876, 2.82799271880609039780699697083, 3.03231072742430237417772370014, 3.86616723306911510718976498766, 4.02353898938027299224793155404, 4.69782393429834785439897408207, 4.75833103936539815706254760731, 5.63016141333174865729898076790, 5.70397031355003125619207722594, 6.21583090682654155982218549237, 6.43505295599236003113349769683, 6.97584779158917003164441571871, 7.24470093947686550662384877534, 7.922065897600999570356155061410, 8.028316523365771278888743096194, 8.660681346588510715851278661209, 8.839020637679438962438054246907