| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 5·5-s + 24·6-s − 14·7-s + 32·8-s + 27·9-s + 20·10-s − 17·11-s + 72·12-s − 26·13-s − 56·14-s + 30·15-s + 80·16-s + 135·17-s + 108·18-s + 171·19-s + 60·20-s − 84·21-s − 68·22-s + 293·23-s + 192·24-s + 119·25-s − 104·26-s + 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 0.632·10-s − 0.465·11-s + 1.73·12-s − 0.554·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 1.92·17-s + 1.41·18-s + 2.06·19-s + 0.670·20-s − 0.872·21-s − 0.658·22-s + 2.65·23-s + 1.63·24-s + 0.951·25-s − 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(15.83310136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.83310136\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - p T - 94 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 17 T + 2384 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 135 T + 14032 T^{2} - 135 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 p T + 20678 T^{2} - 9 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 293 T + 45446 T^{2} - 293 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 193 T + 57740 T^{2} + 193 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 128 T + 41262 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 471 T + 139604 T^{2} + 471 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 216 T + 127090 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 293 T + 152106 T^{2} - 293 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 398 T + 212222 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 228 T + 288334 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 46 T - 206554 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1043 T + 722772 T^{2} - 1043 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 82 T - 137922 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 866 T + 666542 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 389 T + 386808 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 878 T + 1009278 T^{2} + 878 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 192 T - 108110 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 630 T + 1474138 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 404 T + 1776486 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.47742829658348288455356977973, −10.23023936376494699886273289160, −9.637775929206234312117789344618, −9.514753132116763006980557354578, −8.843454660234085403467071728645, −8.436656871391954941294190974523, −7.51919880234155922228137059409, −7.47126590372247331822848443356, −7.04803611643695299576615350791, −6.64296465201033379377416828192, −5.60942026402957979625553889222, −5.52667702358717551566669067649, −5.08451643943948029057707684755, −4.49708888898989823736063886028, −3.55184390675154162695556200882, −3.32965776221302750318487901393, −2.90176869547631116019849731637, −2.53628623031009841771416164013, −1.43760477431384313601420420037, −0.969395545041252444364629945843,
0.969395545041252444364629945843, 1.43760477431384313601420420037, 2.53628623031009841771416164013, 2.90176869547631116019849731637, 3.32965776221302750318487901393, 3.55184390675154162695556200882, 4.49708888898989823736063886028, 5.08451643943948029057707684755, 5.52667702358717551566669067649, 5.60942026402957979625553889222, 6.64296465201033379377416828192, 7.04803611643695299576615350791, 7.47126590372247331822848443356, 7.51919880234155922228137059409, 8.436656871391954941294190974523, 8.843454660234085403467071728645, 9.514753132116763006980557354578, 9.637775929206234312117789344618, 10.23023936376494699886273289160, 10.47742829658348288455356977973
