| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s − 2·7-s − 4·8-s + 8·10-s − 6·11-s + 4·14-s + 5·16-s + 8·17-s + 6·19-s − 12·20-s + 12·22-s + 2·23-s + 5·25-s − 6·28-s + 2·29-s − 4·31-s − 6·32-s − 16·34-s + 8·35-s + 14·37-s − 12·38-s + 16·40-s − 2·41-s − 2·43-s − 18·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s − 0.755·7-s − 1.41·8-s + 2.52·10-s − 1.80·11-s + 1.06·14-s + 5/4·16-s + 1.94·17-s + 1.37·19-s − 2.68·20-s + 2.55·22-s + 0.417·23-s + 25-s − 1.13·28-s + 0.371·29-s − 0.718·31-s − 1.06·32-s − 2.74·34-s + 1.35·35-s + 2.30·37-s − 1.94·38-s + 2.52·40-s − 0.312·41-s − 0.304·43-s − 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 3 p T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 207 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.295142690441896045838102616374, −7.964148992169479860984990627056, −7.927896683684389523089182425908, −7.55448377230103792037925796389, −7.27348662695797111908308120222, −7.07704081481488914219961173888, −6.28269004821249121774789023850, −5.99008302539566747478478349696, −5.42271508172174471658740215124, −5.33078490083814939269586236629, −4.50195461608258171641224450849, −4.21904634530811705282532200219, −3.36429126814305742618972205917, −3.29815734924022765347393987105, −2.84453503914367121914553257580, −2.51942714308908072290373260202, −1.37785165949557012853961381382, −1.06324582863952725990020118708, 0, 0,
1.06324582863952725990020118708, 1.37785165949557012853961381382, 2.51942714308908072290373260202, 2.84453503914367121914553257580, 3.29815734924022765347393987105, 3.36429126814305742618972205917, 4.21904634530811705282532200219, 4.50195461608258171641224450849, 5.33078490083814939269586236629, 5.42271508172174471658740215124, 5.99008302539566747478478349696, 6.28269004821249121774789023850, 7.07704081481488914219961173888, 7.27348662695797111908308120222, 7.55448377230103792037925796389, 7.927896683684389523089182425908, 7.964148992169479860984990627056, 8.295142690441896045838102616374
