| L(s) = 1 | + 16·2-s − 30·3-s − 20·4-s + 250·5-s − 480·6-s − 686·7-s − 2.68e3·8-s − 2.11e3·9-s + 4.00e3·10-s − 7.90e3·11-s + 600·12-s − 1.78e4·13-s − 1.09e4·14-s − 7.50e3·15-s − 2.11e4·16-s − 2.39e3·17-s − 3.38e4·18-s − 3.61e3·19-s − 5.00e3·20-s + 2.05e4·21-s − 1.26e5·22-s + 1.38e4·23-s + 8.06e4·24-s + 4.68e4·25-s − 2.85e5·26-s + 8.82e4·27-s + 1.37e4·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.641·3-s − 0.156·4-s + 0.894·5-s − 0.907·6-s − 0.755·7-s − 1.85·8-s − 0.967·9-s + 1.26·10-s − 1.79·11-s + 0.100·12-s − 2.24·13-s − 1.06·14-s − 0.573·15-s − 1.28·16-s − 0.118·17-s − 1.36·18-s − 0.120·19-s − 0.139·20-s + 0.484·21-s − 2.53·22-s + 0.237·23-s + 1.19·24-s + 3/5·25-s − 3.18·26-s + 0.863·27-s + 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p^{4} T + 69 p^{2} T^{2} - p^{11} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 10 p T + 335 p^{2} T^{2} + 10 p^{8} T^{3} + p^{14} T^{4} \) |
| 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
−14.25088828195067961566883804782, −14.19709681675455236578367004778, −13.44155964499683132262157172551, −12.98908326191473681033799700492, −12.29213353332133251620020554844, −12.25743906086812820399430828680, −11.03722773545208182582551190736, −10.34407340264573230991601713483, −9.650193556651075570337320322616, −9.185285343196466433454397192081, −8.155798047008556568923218970963, −7.17471755688965879825259971910, −6.08785636667174997895236164982, −5.53377260863159059873142106178, −5.11756419308441292598345810012, −4.45354951563359128570893405659, −2.90705316331124143992474431059, −2.62251023264611759355201808733, 0, 0,
2.62251023264611759355201808733, 2.90705316331124143992474431059, 4.45354951563359128570893405659, 5.11756419308441292598345810012, 5.53377260863159059873142106178, 6.08785636667174997895236164982, 7.17471755688965879825259971910, 8.155798047008556568923218970963, 9.185285343196466433454397192081, 9.650193556651075570337320322616, 10.34407340264573230991601713483, 11.03722773545208182582551190736, 12.25743906086812820399430828680, 12.29213353332133251620020554844, 12.98908326191473681033799700492, 13.44155964499683132262157172551, 14.19709681675455236578367004778, 14.25088828195067961566883804782
