| L(s) = 1 | + 2·2-s − 5·4-s − 8·5-s − 28·7-s − 12·8-s − 16·10-s + 72·11-s − 60·13-s − 56·14-s − 11·16-s − 96·17-s − 148·19-s + 40·20-s + 144·22-s − 46·23-s − 74·25-s − 120·26-s + 140·28-s + 204·29-s − 168·31-s − 122·32-s − 192·34-s + 224·35-s + 116·37-s − 296·38-s + 96·40-s − 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/8·4-s − 0.715·5-s − 1.51·7-s − 0.530·8-s − 0.505·10-s + 1.97·11-s − 1.28·13-s − 1.06·14-s − 0.171·16-s − 1.36·17-s − 1.78·19-s + 0.447·20-s + 1.39·22-s − 0.417·23-s − 0.591·25-s − 0.905·26-s + 0.944·28-s + 1.30·29-s − 0.973·31-s − 0.673·32-s − 0.968·34-s + 1.08·35-s + 0.515·37-s − 1.26·38-s + 0.379·40-s − 0.0152·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 9 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−12.06132786428238794874788048828, −11.25678540656775151989507403257, −10.89779754449175077336215185777, −10.12891452228298300512903778709, −9.501259197425938196948591801396, −9.430446786063440349718171838769, −8.723152197346698921612891831737, −8.394959704738237188382798298039, −7.50544644875786313813837304064, −6.85113279139932214816535044586, −6.32788804697468396728078837971, −6.32364180697561043563539853805, −5.07612916495418896418534243292, −4.49051048686462884503053360268, −4.06630158404476451562751306623, −3.69594002042427584732786389401, −2.79370264610506379733462122524, −1.80474675132387779577310287004, 0, 0,
1.80474675132387779577310287004, 2.79370264610506379733462122524, 3.69594002042427584732786389401, 4.06630158404476451562751306623, 4.49051048686462884503053360268, 5.07612916495418896418534243292, 6.32364180697561043563539853805, 6.32788804697468396728078837971, 6.85113279139932214816535044586, 7.50544644875786313813837304064, 8.394959704738237188382798298039, 8.723152197346698921612891831737, 9.430446786063440349718171838769, 9.501259197425938196948591801396, 10.12891452228298300512903778709, 10.89779754449175077336215185777, 11.25678540656775151989507403257, 12.06132786428238794874788048828
