| L(s) = 1 | + 3-s + 2·4-s + 3·5-s + 3·9-s + 2·12-s + 3·15-s + 6·20-s − 36·23-s + 5·25-s + 5·31-s + 6·36-s − 7·37-s + 9·45-s + 12·47-s + 7·49-s − 6·53-s + 15·59-s + 6·60-s + 52·67-s − 36·69-s + 3·71-s + 5·75-s − 36·89-s − 72·92-s + 5·93-s − 17·97-s + 10·100-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s + 1.34·5-s + 9-s + 0.577·12-s + 0.774·15-s + 1.34·20-s − 7.50·23-s + 25-s + 0.898·31-s + 36-s − 1.15·37-s + 1.34·45-s + 1.75·47-s + 49-s − 0.824·53-s + 1.95·59-s + 0.774·60-s + 6.35·67-s − 4.33·69-s + 0.356·71-s + 0.577·75-s − 3.81·89-s − 7.50·92-s + 0.518·93-s − 1.72·97-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998172524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998172524\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 185 p T^{5} - 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 7 T + 12 T^{2} - 175 T^{3} - 1669 T^{4} - 175 p T^{5} + 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 600 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 1605 p T^{5} + 166 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 + 17 T + 192 T^{2} + 1615 T^{3} + 8831 T^{4} + 1615 p T^{5} + 192 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.865733517318595438002667835754, −9.704986785959123194140910117979, −9.571647589286248238059675020654, −9.396652134805677791746800491407, −8.698767798797060376802557990818, −8.285793286391824580858809377588, −8.147067558637862565197103207270, −7.986189174377125712027807869998, −7.973924919721063294100172603837, −7.10172663489530703404904093521, −6.91888483306900021544764337416, −6.85246772846024821651366745207, −6.35840096242506855460912646301, −5.92954515821149489802277856239, −5.92951760919670959488528225590, −5.43391847772256698767773231700, −5.34629845659035211590577158137, −4.44527678217534267039077911646, −4.00059019530503744928279230377, −3.85811350682600829681631682069, −3.80119557959838207945530827534, −2.54351762753205675592695845243, −2.32782473582246797319267446190, −2.06610799285378510264236344863, −1.71651443117336284010436109300,
1.71651443117336284010436109300, 2.06610799285378510264236344863, 2.32782473582246797319267446190, 2.54351762753205675592695845243, 3.80119557959838207945530827534, 3.85811350682600829681631682069, 4.00059019530503744928279230377, 4.44527678217534267039077911646, 5.34629845659035211590577158137, 5.43391847772256698767773231700, 5.92951760919670959488528225590, 5.92954515821149489802277856239, 6.35840096242506855460912646301, 6.85246772846024821651366745207, 6.91888483306900021544764337416, 7.10172663489530703404904093521, 7.973924919721063294100172603837, 7.986189174377125712027807869998, 8.147067558637862565197103207270, 8.285793286391824580858809377588, 8.698767798797060376802557990818, 9.396652134805677791746800491407, 9.571647589286248238059675020654, 9.704986785959123194140910117979, 9.865733517318595438002667835754
