| L(s) = 1 | + 28·3-s − 42·5-s + 211·9-s − 660·11-s + 644·13-s − 1.17e3·15-s + 210·17-s + 3.72e3·19-s − 24·23-s − 1.00e3·25-s − 3.33e3·27-s + 5.53e3·29-s + 2.80e3·31-s − 1.84e4·33-s − 1.32e4·37-s + 1.80e4·39-s − 4.11e3·41-s − 1.34e4·43-s − 8.86e3·45-s + 8.06e3·47-s + 5.88e3·51-s − 5.39e4·53-s + 2.77e4·55-s + 1.04e5·57-s + 3.60e4·59-s − 8.39e4·61-s − 2.70e4·65-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.751·5-s + 0.868·9-s − 1.64·11-s + 1.05·13-s − 1.34·15-s + 0.176·17-s + 2.36·19-s − 0.00946·23-s − 0.320·25-s − 0.879·27-s + 1.22·29-s + 0.523·31-s − 2.95·33-s − 1.58·37-s + 1.89·39-s − 0.382·41-s − 1.10·43-s − 0.652·45-s + 0.532·47-s + 0.316·51-s − 2.63·53-s + 1.23·55-s + 4.25·57-s + 1.34·59-s − 2.88·61-s − 0.794·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 28 T + 191 p T^{2} - 28 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 42 T + 2767 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 60 p T + 382933 T^{2} + 60 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 644 T + 595134 T^{2} - 644 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 210 T + 2787955 T^{2} - 210 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 196 p T + 8410413 T^{2} - 196 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 24 T + 10517449 T^{2} + 24 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5532 T + 36367390 T^{2} - 5532 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2800 T + 38019873 T^{2} - 2800 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13238 T + 173077551 T^{2} + 13238 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4116 T + 163369462 T^{2} + 4116 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6716 T + p^{5} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8064 T - 63701423 T^{2} - 8064 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 53958 T + 1531762783 T^{2} + 53958 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 36036 T + 1411852261 T^{2} - 36036 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 83986 T + 3450528855 T^{2} + 83986 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2660 T + 2661593085 T^{2} - 2660 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1008 p T + 3116937742 T^{2} + 1008 p^{6} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 31318 T + 997479867 T^{2} + 31318 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 51136 T + 1977429681 T^{2} + 51136 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6216 T + 6437179414 T^{2} - 6216 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 166278 T + 17776949803 T^{2} + 166278 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8260 T + 11911603014 T^{2} + 8260 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.139165226575988505527927282882, −8.836011721636142305957217887534, −8.304939440282318683813272879000, −8.162913273494487508703673340255, −7.75244941698651743035443226954, −7.49557710289560812454170785488, −6.94052033843498701832193508870, −6.33795826832038386607955291570, −5.61118146911858915184983210969, −5.37234396090611504748137970981, −4.75456554287692472853465079940, −4.21114134310529893649991186834, −3.38919731333108019090824196901, −3.31531713988108530776069415024, −2.85774593339162307987703426699, −2.55299351929761451792048409165, −1.41154210407477838249747041414, −1.37994708492961811316041756148, 0, 0,
1.37994708492961811316041756148, 1.41154210407477838249747041414, 2.55299351929761451792048409165, 2.85774593339162307987703426699, 3.31531713988108530776069415024, 3.38919731333108019090824196901, 4.21114134310529893649991186834, 4.75456554287692472853465079940, 5.37234396090611504748137970981, 5.61118146911858915184983210969, 6.33795826832038386607955291570, 6.94052033843498701832193508870, 7.49557710289560812454170785488, 7.75244941698651743035443226954, 8.162913273494487508703673340255, 8.304939440282318683813272879000, 8.836011721636142305957217887534, 9.139165226575988505527927282882
