| L(s) = 1 | − 16·4-s + 120·11-s + 256·16-s − 1.91e3·19-s + 1.11e4·29-s − 7.18e3·31-s − 3.83e4·41-s − 1.92e3·44-s + 2.63e3·49-s + 5.26e4·59-s − 6.21e4·61-s − 4.09e3·64-s − 1.22e4·71-s + 3.05e4·76-s − 1.48e5·79-s − 6.54e4·89-s + 4.40e4·101-s + 1.70e4·109-s − 1.78e5·116-s − 3.11e5·121-s + 1.14e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.299·11-s + 1/4·16-s − 1.21·19-s + 2.46·29-s − 1.34·31-s − 3.56·41-s − 0.149·44-s + 0.156·49-s + 1.97·59-s − 2.13·61-s − 1/8·64-s − 0.288·71-s + 0.607·76-s − 2.68·79-s − 0.876·89-s + 0.429·101-s + 0.137·109-s − 1.23·116-s − 1.93·121-s + 0.671·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1647146016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1647146016\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 2638 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 60 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 309622 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2668318 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 956 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12512686 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5574 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3592 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 67150150 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 19194 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 116701030 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 71387614 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 141171770 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 26340 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 31090 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2417875798 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3492931822 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 74408 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7836246262 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 32742 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10408550210 T^{2} + p^{10} T^{4} \) |
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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
−10.76338165471746624580506418690, −10.11331200178994557150058408981, −9.736340261899489014752228778938, −8.916212591585891512448789302863, −8.776506177212894162431892647298, −8.361062556335357317445858470030, −7.954729427233739024662154568846, −7.11941035391945877398894704064, −6.88333211029644580372838160426, −6.32969307137263377534888027910, −5.86716610419857470424684764600, −5.10288064414031040979642756947, −4.85438675428714799650903514674, −4.19554846174002399751199244839, −3.72296340769228336339451732452, −3.07925598898133590186809360902, −2.47040574690854360132459868575, −1.63678104265634774763270944253, −1.18395069004212450496864571581, −0.10134902057674442990477262355,
0.10134902057674442990477262355, 1.18395069004212450496864571581, 1.63678104265634774763270944253, 2.47040574690854360132459868575, 3.07925598898133590186809360902, 3.72296340769228336339451732452, 4.19554846174002399751199244839, 4.85438675428714799650903514674, 5.10288064414031040979642756947, 5.86716610419857470424684764600, 6.32969307137263377534888027910, 6.88333211029644580372838160426, 7.11941035391945877398894704064, 7.954729427233739024662154568846, 8.361062556335357317445858470030, 8.776506177212894162431892647298, 8.916212591585891512448789302863, 9.736340261899489014752228778938, 10.11331200178994557150058408981, 10.76338165471746624580506418690
