L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 4·7-s − 4·8-s + 3·9-s − 6·11-s − 6·12-s − 3·13-s − 8·14-s + 5·16-s + 9·17-s − 6·18-s − 10·19-s − 8·21-s + 12·22-s + 12·23-s + 8·24-s + 6·26-s − 4·27-s + 12·28-s − 5·29-s − 6·31-s − 6·32-s + 12·33-s − 18·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.51·7-s − 1.41·8-s + 9-s − 1.80·11-s − 1.73·12-s − 0.832·13-s − 2.13·14-s + 5/4·16-s + 2.18·17-s − 1.41·18-s − 2.29·19-s − 1.74·21-s + 2.55·22-s + 2.50·23-s + 1.63·24-s + 1.17·26-s − 0.769·27-s + 2.26·28-s − 0.928·29-s − 1.07·31-s − 1.06·32-s + 2.08·33-s − 3.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14062500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14062500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 177 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 183 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 293 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−8.151982306961392511014426216207, −7.997055380776331572648735866953, −7.52956833677045865887194948170, −7.46133695483654061657086677844, −6.95334655871307471255067266908, −6.78070098886909779844989583588, −5.94565930321878912176040031756, −5.74751437804703587918797517088, −5.26045632940998153227814077550, −5.22939497959559114773746464413, −4.61914620406100735710287348611, −4.39037955941688320716652101337, −3.42111005767575422146274936702, −3.17617963695677721852284600107, −2.37525957408376665510133714282, −2.15605174263596132333072658192, −1.34906812509579783043040958521, −1.23701953300911402566996408412, 0, 0,
1.23701953300911402566996408412, 1.34906812509579783043040958521, 2.15605174263596132333072658192, 2.37525957408376665510133714282, 3.17617963695677721852284600107, 3.42111005767575422146274936702, 4.39037955941688320716652101337, 4.61914620406100735710287348611, 5.22939497959559114773746464413, 5.26045632940998153227814077550, 5.74751437804703587918797517088, 5.94565930321878912176040031756, 6.78070098886909779844989583588, 6.95334655871307471255067266908, 7.46133695483654061657086677844, 7.52956833677045865887194948170, 7.997055380776331572648735866953, 8.151982306961392511014426216207