| L(s) = 1 | − 16·2-s − 20·4-s − 250·5-s − 686·7-s + 2.68e3·8-s + 4.00e3·10-s + 7.90e3·11-s − 1.78e4·13-s + 1.09e4·14-s − 2.11e4·16-s + 2.39e3·17-s − 3.61e3·19-s + 5.00e3·20-s − 1.26e5·22-s − 1.38e4·23-s + 4.68e4·25-s + 2.85e5·26-s + 1.37e4·28-s + 1.26e5·29-s + 2.52e5·31-s − 1.01e5·32-s − 3.83e4·34-s + 1.71e5·35-s − 2.65e5·37-s + 5.77e4·38-s − 6.72e5·40-s + 1.11e5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.156·4-s − 0.894·5-s − 0.755·7-s + 1.85·8-s + 1.26·10-s + 1.79·11-s − 2.24·13-s + 1.06·14-s − 1.28·16-s + 0.118·17-s − 0.120·19-s + 0.139·20-s − 2.53·22-s − 0.237·23-s + 3/5·25-s + 3.18·26-s + 0.118·28-s + 0.966·29-s + 1.52·31-s − 0.545·32-s − 0.167·34-s + 0.676·35-s − 0.862·37-s + 0.170·38-s − 1.66·40-s + 0.253·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p^{4} T + 69 p^{2} T^{2} + p^{11} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7906 T + 48246951 T^{2} - 7906 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 17818 T + 197179459 T^{2} + 17818 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2398 T + 607214547 T^{2} - 2398 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3612 T + 439599498 T^{2} + 3612 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13844 T + 1829416578 T^{2} + 13844 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 126898 T + 13660776923 T^{2} - 126898 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 252768 T + 48616095374 T^{2} - 252768 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 265860 T + 123746037230 T^{2} + 265860 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 111920 T + 331245931458 T^{2} - 111920 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 947572 T + 741530960314 T^{2} - 947572 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 271274 T + 889978835879 T^{2} + 271274 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1267792 T + 2564062930746 T^{2} - 1267792 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1360120 T + 4158592150838 T^{2} - 1360120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1813680 T + 7102192237178 T^{2} + 1813680 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2189312 T + 2038788130198 T^{2} + 2189312 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1494928 T + 16579508918702 T^{2} - 1494928 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7169788 T + 32961331353526 T^{2} - 7169788 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7942974 T + 52633585962783 T^{2} + 7942974 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 304712 T + 15462893048006 T^{2} - 304712 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17943528 T + 166259116359218 T^{2} - 17943528 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4258074 T + 145736945716539 T^{2} - 4258074 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−9.856916704432257487726368606957, −9.762026681107792349655995821107, −9.162688733891007829443087212879, −9.074399932175479723740283592213, −8.218127872632954978929908046839, −8.176744457178940515688776637855, −7.25774702023086310329099941795, −7.23733247717848268060315894348, −6.55973576397895980395111533977, −6.00273004347093229344625400815, −4.89929784950515326110686431240, −4.80061133198788981019337012547, −3.99559834360250094633011893998, −3.80193208114921394573342180703, −2.78193649676620490820640823447, −2.27866657053432208330529432132, −1.04774322197914892361878349132, −1.01368303051723173992798088275, 0, 0,
1.01368303051723173992798088275, 1.04774322197914892361878349132, 2.27866657053432208330529432132, 2.78193649676620490820640823447, 3.80193208114921394573342180703, 3.99559834360250094633011893998, 4.80061133198788981019337012547, 4.89929784950515326110686431240, 6.00273004347093229344625400815, 6.55973576397895980395111533977, 7.23733247717848268060315894348, 7.25774702023086310329099941795, 8.176744457178940515688776637855, 8.218127872632954978929908046839, 9.074399932175479723740283592213, 9.162688733891007829443087212879, 9.762026681107792349655995821107, 9.856916704432257487726368606957
