L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s + 4·8-s + 3·9-s + 4·10-s + 6·11-s + 6·12-s + 4·15-s + 5·16-s + 8·17-s + 6·18-s + 2·19-s + 6·20-s + 12·22-s + 8·24-s − 2·25-s + 4·27-s + 8·30-s + 4·31-s + 6·32-s + 12·33-s + 16·34-s + 9·36-s − 18·37-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s + 1.80·11-s + 1.73·12-s + 1.03·15-s + 5/4·16-s + 1.94·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s + 2.55·22-s + 1.63·24-s − 2/5·25-s + 0.769·27-s + 1.46·30-s + 0.718·31-s + 1.06·32-s + 2.08·33-s + 2.74·34-s + 3/2·36-s − 2.95·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10074276 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10074276 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(19.65790953\) |
\(L(\frac12)\) |
\(\approx\) |
\(19.65790953\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18 T + 150 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.961962908553550004489310627762, −8.646515706565953878841644923077, −7.88077482279980279661801468170, −7.68009652823819855391609531230, −7.31049354114779203849974613491, −7.00740905512036870265019794104, −6.35508843535325978501003240906, −6.33977126711634566005120384806, −5.62210841135063879684587788492, −5.58681294003665807559192020104, −5.00481801069111586271087232843, −4.54918994635621799668434980638, −3.96099586854095752007575502368, −3.75483423974099476760505675854, −3.43654483710125286533570535496, −2.91139738691773074664742373955, −2.51716272798110563496820644548, −1.89303614735994328170303462046, −1.39414480973934441384358648368, −1.15059298584104255275817229374,
1.15059298584104255275817229374, 1.39414480973934441384358648368, 1.89303614735994328170303462046, 2.51716272798110563496820644548, 2.91139738691773074664742373955, 3.43654483710125286533570535496, 3.75483423974099476760505675854, 3.96099586854095752007575502368, 4.54918994635621799668434980638, 5.00481801069111586271087232843, 5.58681294003665807559192020104, 5.62210841135063879684587788492, 6.33977126711634566005120384806, 6.35508843535325978501003240906, 7.00740905512036870265019794104, 7.31049354114779203849974613491, 7.68009652823819855391609531230, 7.88077482279980279661801468170, 8.646515706565953878841644923077, 8.961962908553550004489310627762