| L(s) = 1 | − 9·2-s + 19·4-s + 108·7-s + 99·8-s − 168·11-s + 1.29e3·13-s − 972·14-s − 501·16-s − 576·17-s − 1.33e3·19-s + 1.51e3·22-s − 5.90e3·23-s − 1.16e4·26-s + 2.05e3·28-s + 3.55e3·29-s − 1.16e4·31-s + 2.11e3·32-s + 5.18e3·34-s + 1.46e4·37-s + 1.20e4·38-s − 1.81e3·41-s − 7.56e3·43-s − 3.19e3·44-s + 5.31e4·46-s − 3.24e3·47-s − 1.20e4·49-s + 2.46e4·52-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.593·4-s + 0.833·7-s + 0.546·8-s − 0.418·11-s + 2.12·13-s − 1.32·14-s − 0.489·16-s − 0.483·17-s − 0.849·19-s + 0.666·22-s − 2.32·23-s − 3.38·26-s + 0.494·28-s + 0.784·29-s − 2.17·31-s + 0.365·32-s + 0.769·34-s + 1.76·37-s + 1.35·38-s − 0.168·41-s − 0.623·43-s − 0.248·44-s + 3.70·46-s − 0.213·47-s − 0.716·49-s + 1.26·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 9 T + 31 p T^{2} + 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 108 T + 23714 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 168 T + 69634 T^{2} + 168 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1296 T + 1005494 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 576 T + 2486558 T^{2} + 576 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1336 T + 5283078 T^{2} + 1336 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5904 T + 21414686 T^{2} + 5904 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3552 T + 27566938 T^{2} - 3552 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11648 T + 90715902 T^{2} + 11648 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14688 T + 189133094 T^{2} - 14688 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1812 T + 29066422 T^{2} + 1812 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7560 T + 268114310 T^{2} + 7560 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3240 T + 424402910 T^{2} + 3240 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 22176 T + 869263526 T^{2} - 22176 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 57336 T + 2002300258 T^{2} + 57336 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 30140 T + 1692991518 T^{2} + 30140 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5184 T + 2244311078 T^{2} + 5184 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17424 T + 3450786046 T^{2} + 17424 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3456 T + 4135172546 T^{2} + 3456 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 57520 T + 5031936798 T^{2} + 57520 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 133992 T + 11327999318 T^{2} + 133992 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 136764 T + 15341778358 T^{2} + 136764 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 85536 T + 13589022338 T^{2} - 85536 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.88414662052391397458986740532, −10.73298146231551014745995037170, −9.963619484676334322063789464212, −9.730984438326699188906291403128, −8.940453701050878005934934202023, −8.725124289078788148757756497181, −8.184203687013955763833645375925, −8.041552976008442977966934577074, −7.44473974548268264779038320415, −6.61339758031697440414813105037, −5.92827324892733344171787470363, −5.76247998688122225187561427516, −4.57714444208178300985588655361, −4.24327159212855601201099114932, −3.54641969579040297756386438735, −2.50676054631271740113353845210, −1.48776007009664375402752041893, −1.40180445340640330753247274043, 0, 0,
1.40180445340640330753247274043, 1.48776007009664375402752041893, 2.50676054631271740113353845210, 3.54641969579040297756386438735, 4.24327159212855601201099114932, 4.57714444208178300985588655361, 5.76247998688122225187561427516, 5.92827324892733344171787470363, 6.61339758031697440414813105037, 7.44473974548268264779038320415, 8.041552976008442977966934577074, 8.184203687013955763833645375925, 8.725124289078788148757756497181, 8.940453701050878005934934202023, 9.730984438326699188906291403128, 9.963619484676334322063789464212, 10.73298146231551014745995037170, 10.88414662052391397458986740532
