| L(s) = 1 | + 2·2-s − 3-s + 3·4-s + 4·5-s − 2·6-s + 3·7-s + 4·8-s − 9-s + 8·10-s + 2·11-s − 3·12-s + 3·13-s + 6·14-s − 4·15-s + 5·16-s + 3·17-s − 2·18-s + 12·20-s − 3·21-s + 4·22-s + 7·23-s − 4·24-s + 2·25-s + 6·26-s + 9·28-s − 29-s − 8·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s + 1.78·5-s − 0.816·6-s + 1.13·7-s + 1.41·8-s − 1/3·9-s + 2.52·10-s + 0.603·11-s − 0.866·12-s + 0.832·13-s + 1.60·14-s − 1.03·15-s + 5/4·16-s + 0.727·17-s − 0.471·18-s + 2.68·20-s − 0.654·21-s + 0.852·22-s + 1.45·23-s − 0.816·24-s + 2/5·25-s + 1.17·26-s + 1.70·28-s − 0.185·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63075364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63075364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.49046773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.49046773\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 98 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 186 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.76149139709409670564370199030, −7.49656327108103711519974684412, −7.30739420095091067637585896249, −6.77258882591034770576810068363, −6.26152210085320616205618021448, −6.20922550054195113709633633278, −5.87130281168689492098727740929, −5.58312156423369932347845145021, −5.20909408360791196359647595809, −5.05917114168770782977842868853, −4.64430177490141412929841993326, −4.16835199699085956052137885388, −3.68183590469577613779774575966, −3.51966767263986573898780279740, −2.72051812345261325163503029859, −2.66834210528100327403904088613, −1.84362520720394966377157797982, −1.73505123945761296776573171653, −1.32574849154556456165089571751, −0.72496175532024712643623898876,
0.72496175532024712643623898876, 1.32574849154556456165089571751, 1.73505123945761296776573171653, 1.84362520720394966377157797982, 2.66834210528100327403904088613, 2.72051812345261325163503029859, 3.51966767263986573898780279740, 3.68183590469577613779774575966, 4.16835199699085956052137885388, 4.64430177490141412929841993326, 5.05917114168770782977842868853, 5.20909408360791196359647595809, 5.58312156423369932347845145021, 5.87130281168689492098727740929, 6.20922550054195113709633633278, 6.26152210085320616205618021448, 6.77258882591034770576810068363, 7.30739420095091067637585896249, 7.49656327108103711519974684412, 7.76149139709409670564370199030
