| L(s) = 1 | + 2·2-s + 6·3-s − 5·4-s + 12·6-s + 28·7-s − 12·8-s + 27·9-s − 72·11-s − 30·12-s + 60·13-s + 56·14-s − 11·16-s − 96·17-s + 54·18-s − 148·19-s + 168·21-s − 144·22-s − 46·23-s − 72·24-s + 120·26-s + 108·27-s − 140·28-s − 204·29-s − 168·31-s − 122·32-s − 432·33-s − 192·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 5/8·4-s + 0.816·6-s + 1.51·7-s − 0.530·8-s + 9-s − 1.97·11-s − 0.721·12-s + 1.28·13-s + 1.06·14-s − 0.171·16-s − 1.36·17-s + 0.707·18-s − 1.78·19-s + 1.74·21-s − 1.39·22-s − 0.417·23-s − 0.612·24-s + 0.905·26-s + 0.769·27-s − 0.944·28-s − 1.30·29-s − 0.973·31-s − 0.673·32-s − 2.27·33-s − 0.968·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 9 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 p T + 874 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 72 T + 3566 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 60 T + 5006 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 96 T + 11930 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 148 T + 14586 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 204 T + 24334 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 168 T + 43310 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 116 T + 89878 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 132438 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 420 T + 200522 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 48 T + 146270 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 32 T + 297210 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 40 T + 211446 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 764 T + 580678 T^{2} + 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 988 T + 675034 T^{2} - 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 224 T + 73998 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 820 T + 921046 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1772 T + 1743226 T^{2} - 1772 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1480 T + 1335006 T^{2} + 1480 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 744 T + 1044314 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 260 T + 178758 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.659374696639208729249847036764, −8.499371982951712201977177384550, −7.87856474228485645218249280336, −7.80612813662392544262441093964, −7.42443576673029307016418393918, −6.79599050415356389447333356503, −6.22548528264916967001140814326, −5.88603932685045812321793255650, −5.15096171190085313178987284286, −5.09959200269593206745185160073, −4.42447433356078448436677085926, −4.34108341644977490859426905557, −3.70327090202665127754689102481, −3.48022220320783984611284976958, −2.54426258077692497859776841854, −2.21539106701707074549748030542, −1.90582980938998709478424543111, −1.26094946157281827623521827876, 0, 0,
1.26094946157281827623521827876, 1.90582980938998709478424543111, 2.21539106701707074549748030542, 2.54426258077692497859776841854, 3.48022220320783984611284976958, 3.70327090202665127754689102481, 4.34108341644977490859426905557, 4.42447433356078448436677085926, 5.09959200269593206745185160073, 5.15096171190085313178987284286, 5.88603932685045812321793255650, 6.22548528264916967001140814326, 6.79599050415356389447333356503, 7.42443576673029307016418393918, 7.80612813662392544262441093964, 7.87856474228485645218249280336, 8.499371982951712201977177384550, 8.659374696639208729249847036764
