| L(s) = 1 | + 4·2-s + 12·4-s − 7·5-s − 14·7-s + 32·8-s − 28·10-s + 22·11-s − 27·13-s − 56·14-s + 80·16-s + 64·17-s − 117·19-s − 84·20-s + 88·22-s + 60·23-s − 159·25-s − 108·26-s − 168·28-s + 135·29-s − 134·31-s + 192·32-s + 256·34-s + 98·35-s − 257·37-s − 468·38-s − 224·40-s − 86·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.626·5-s − 0.755·7-s + 1.41·8-s − 0.885·10-s + 0.603·11-s − 0.576·13-s − 1.06·14-s + 5/4·16-s + 0.913·17-s − 1.41·19-s − 0.939·20-s + 0.852·22-s + 0.543·23-s − 1.27·25-s − 0.814·26-s − 1.13·28-s + 0.864·29-s − 0.776·31-s + 1.06·32-s + 1.29·34-s + 0.473·35-s − 1.14·37-s − 1.99·38-s − 0.885·40-s − 0.327·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 7 T + 208 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 27 T + 3220 T^{2} + 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 64 T + 3038 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 117 T + 12746 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 60 T + 3534 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 135 T + 50676 T^{2} - 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 134 T + 63854 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 257 T + 83912 T^{2} + 257 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 86 T + 61354 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 208 T + 138582 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 227 T + 216134 T^{2} + 227 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 378 T + 331522 T^{2} + 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 371 T + 314914 T^{2} + 371 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 872 T + 643190 T^{2} + 872 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 269 T + 390410 T^{2} - 269 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 100 T - 11666 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1493 T + 1328732 T^{2} + 1493 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1260 T + 1187678 T^{2} + 1260 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 662 T + 1247710 T^{2} - 662 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 80 T + 1306510 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1666 T + 2501658 T^{2} + 1666 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.768205707711305126504133860633, −8.745296484620261026289451515251, −8.010577437502583259677500850882, −7.64036623929727495002717417402, −7.31556269137395177537962688711, −6.82117307752201127863986261732, −6.30744158614150512958181724138, −6.30524642740275702524632314902, −5.55757004441918442855547358598, −5.26855207308206549885416636342, −4.58908942350582336579928715741, −4.37776381302238348181523554066, −3.78921181877941095778456847783, −3.46610426546915934414109971132, −2.98594033609210722322299947837, −2.54570856865060824307384016065, −1.66968165604191719900013650532, −1.40738328260754764904493670406, 0, 0,
1.40738328260754764904493670406, 1.66968165604191719900013650532, 2.54570856865060824307384016065, 2.98594033609210722322299947837, 3.46610426546915934414109971132, 3.78921181877941095778456847783, 4.37776381302238348181523554066, 4.58908942350582336579928715741, 5.26855207308206549885416636342, 5.55757004441918442855547358598, 6.30524642740275702524632314902, 6.30744158614150512958181724138, 6.82117307752201127863986261732, 7.31556269137395177537962688711, 7.64036623929727495002717417402, 8.010577437502583259677500850882, 8.745296484620261026289451515251, 8.768205707711305126504133860633
