| L(s) = 1 | − 50·5-s − 52·7-s + 560·11-s + 1.38e3·13-s − 148·17-s + 1.00e3·19-s − 2.45e3·23-s + 1.87e3·25-s − 1.34e3·29-s + 2.24e3·31-s + 2.60e3·35-s − 5.94e3·37-s − 2.30e4·41-s − 1.76e4·43-s − 2.90e3·47-s − 2.69e4·49-s + 5.41e3·53-s − 2.80e4·55-s + 6.25e4·59-s + 1.41e4·61-s − 6.94e4·65-s + 8.54e4·67-s + 4.72e4·71-s − 6.74e4·73-s − 2.91e4·77-s + 6.59e4·79-s + 1.08e5·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.401·7-s + 1.39·11-s + 2.27·13-s − 0.124·17-s + 0.635·19-s − 0.966·23-s + 3/5·25-s − 0.295·29-s + 0.420·31-s + 0.358·35-s − 0.713·37-s − 2.14·41-s − 1.45·43-s − 0.192·47-s − 1.60·49-s + 0.264·53-s − 1.24·55-s + 2.34·59-s + 0.485·61-s − 2.03·65-s + 2.32·67-s + 1.11·71-s − 1.48·73-s − 0.559·77-s + 1.18·79-s + 1.73·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.568342546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568342546\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 52 T + 29646 T^{2} + 52 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1388 T + 1149918 T^{2} - 1388 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 148 T - 795706 T^{2} + 148 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1000 T + 4533462 T^{2} - 1000 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2452 T + 6569198 T^{2} + 2452 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2248 T + 57017022 T^{2} - 2248 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5940 T + 123434318 T^{2} + 5940 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 23076 T + 352280470 T^{2} + 23076 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17684 T + 312898614 T^{2} + 17684 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2908 T + 56660030 T^{2} + 2908 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5412 T + 693247822 T^{2} - 5412 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 62584 T + 2277965606 T^{2} - 62584 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 85412 T + 4371910566 T^{2} - 85412 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 47208 T + 4011779662 T^{2} - 47208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 65904 T + 3994274078 T^{2} - 65904 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 108724 T + 10572459494 T^{2} - 108724 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 55020 T + 10818978262 T^{2} - 55020 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.704719138391626311912566461360, −9.558231092951969392762027878280, −8.730755667115549615916677010986, −8.685152811573365968711515932327, −8.098964135258472102089649414937, −8.004157725220559468593713130648, −7.05221671953603613736669626367, −6.74127870402496888699261072675, −6.42853755015557943708057951868, −6.10837302477779153570364515610, −5.20405706781451050498532255341, −5.07885429493266062411514021931, −4.03139355990114457127833519159, −3.89399362627151001755098480783, −3.39856027263166861999511131821, −3.21705653086219638303971498036, −1.92191569206915333981724482023, −1.68514736068514509936462472991, −0.830344355230170535350590162482, −0.52536267852425195496422192860,
0.52536267852425195496422192860, 0.830344355230170535350590162482, 1.68514736068514509936462472991, 1.92191569206915333981724482023, 3.21705653086219638303971498036, 3.39856027263166861999511131821, 3.89399362627151001755098480783, 4.03139355990114457127833519159, 5.07885429493266062411514021931, 5.20405706781451050498532255341, 6.10837302477779153570364515610, 6.42853755015557943708057951868, 6.74127870402496888699261072675, 7.05221671953603613736669626367, 8.004157725220559468593713130648, 8.098964135258472102089649414937, 8.685152811573365968711515932327, 8.730755667115549615916677010986, 9.558231092951969392762027878280, 9.704719138391626311912566461360
