| L(s) = 1 | + 8·2-s + 9·3-s + 48·4-s + 73·5-s + 72·6-s + 155·7-s + 256·8-s − 213·9-s + 584·10-s − 220·11-s + 432·12-s − 338·13-s + 1.24e3·14-s + 657·15-s + 1.28e3·16-s − 189·17-s − 1.70e3·18-s − 2.49e3·19-s + 3.50e3·20-s + 1.39e3·21-s − 1.76e3·22-s − 3.04e3·23-s + 2.30e3·24-s − 343·25-s − 2.70e3·26-s − 2.37e3·27-s + 7.44e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 1.30·5-s + 0.816·6-s + 1.19·7-s + 1.41·8-s − 0.876·9-s + 1.84·10-s − 0.548·11-s + 0.866·12-s − 0.554·13-s + 1.69·14-s + 0.753·15-s + 5/4·16-s − 0.158·17-s − 1.23·18-s − 1.58·19-s + 1.95·20-s + 0.690·21-s − 0.775·22-s − 1.19·23-s + 0.816·24-s − 0.109·25-s − 0.784·26-s − 0.627·27-s + 1.79·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.169054178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.169054178\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - p^{2} T + 98 p T^{2} - p^{7} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 73 T + 5672 T^{2} - 73 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 155 T + 3204 p T^{2} - 155 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 20 p T + 211946 T^{2} + 20 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 189 T - 683408 T^{2} + 189 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2496 T + 5745602 T^{2} + 2496 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3044 T + 12713486 T^{2} + 3044 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1900 T + 27131822 T^{2} - 1900 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2798 T + 28888374 T^{2} - 2798 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 17805 T + 211292840 T^{2} - 17805 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11634 T + 220246402 T^{2} - 11634 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4069 T + 287976354 T^{2} + 4069 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 25489 T + 586297988 T^{2} + 25489 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4614 T + 13331162 p T^{2} + 4614 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 23420 T + 1522838282 T^{2} + 23420 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 96830 T + 3998904378 T^{2} - 96830 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 72440 T + 3995970258 T^{2} - 72440 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 36679 T + 1745469740 T^{2} + 36679 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 47152 T + 2683249326 T^{2} - 47152 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 52024 T + 6594049326 T^{2} - 52024 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 37758 T + 7961553766 T^{2} + 37758 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 105072 T + 12240253294 T^{2} + 105072 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 104392 T + 12944906574 T^{2} - 104392 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−16.65393141509281787789022351935, −15.95103027360583875772429883497, −14.96625483007129307391014044730, −14.75021561878678320725060837236, −14.05199997379584470331190154786, −13.94072404089799965739505584817, −12.99139502186376200844389745737, −12.67548542762211480037290975431, −11.46156167152171366789830604601, −11.28212664178934284505791888180, −10.22118613187508597569152732301, −9.610055249400192744031544780096, −8.229247391209956270999781472236, −8.041958211122648473035761903482, −6.54440263614700288038264891485, −5.87771978695044039499789698161, −5.09509794415253796088132261534, −4.18187615851047687427026452248, −2.55064058072973037014890620831, −2.06394894328285352622557765711,
2.06394894328285352622557765711, 2.55064058072973037014890620831, 4.18187615851047687427026452248, 5.09509794415253796088132261534, 5.87771978695044039499789698161, 6.54440263614700288038264891485, 8.041958211122648473035761903482, 8.229247391209956270999781472236, 9.610055249400192744031544780096, 10.22118613187508597569152732301, 11.28212664178934284505791888180, 11.46156167152171366789830604601, 12.67548542762211480037290975431, 12.99139502186376200844389745737, 13.94072404089799965739505584817, 14.05199997379584470331190154786, 14.75021561878678320725060837236, 14.96625483007129307391014044730, 15.95103027360583875772429883497, 16.65393141509281787789022351935
