| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 8-s − 8·13-s − 14-s − 16-s − 3·17-s − 2·19-s + 21-s − 3·23-s + 24-s − 8·26-s + 27-s + 31-s − 3·34-s + 10·37-s − 2·38-s + 8·39-s − 18·41-s + 42-s − 20·43-s − 3·46-s + 3·47-s + 48-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2.21·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.218·21-s − 0.625·23-s + 0.204·24-s − 1.56·26-s + 0.192·27-s + 0.179·31-s − 0.514·34-s + 1.64·37-s − 0.324·38-s + 1.28·39-s − 2.81·41-s + 0.154·42-s − 3.04·43-s − 0.442·46-s + 0.437·47-s + 0.144·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3881381322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3881381322\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.21335969402079982410963072962, −9.730538780379656371421190911888, −9.614504355951646771465534462245, −8.667591786671410119793608116262, −8.618310615937438547284024461429, −8.060266459254038073469392377787, −7.48187883509705534468151499347, −6.97629222482906056024192964112, −6.78683655162858056919488096929, −6.22854308036208720534033800373, −5.88724220732158892434040578994, −5.17594272644048480983069951448, −4.92021958880353933185533020363, −4.64066406847561342282695939533, −4.05280261251976099551174012308, −3.39627365683952838695544182477, −2.89938881670955343353733750697, −2.28816009574158558705797445878, −1.70702419671036469922504204819, −0.24439565688008791105254155447,
0.24439565688008791105254155447, 1.70702419671036469922504204819, 2.28816009574158558705797445878, 2.89938881670955343353733750697, 3.39627365683952838695544182477, 4.05280261251976099551174012308, 4.64066406847561342282695939533, 4.92021958880353933185533020363, 5.17594272644048480983069951448, 5.88724220732158892434040578994, 6.22854308036208720534033800373, 6.78683655162858056919488096929, 6.97629222482906056024192964112, 7.48187883509705534468151499347, 8.060266459254038073469392377787, 8.618310615937438547284024461429, 8.667591786671410119793608116262, 9.614504355951646771465534462245, 9.730538780379656371421190911888, 10.21335969402079982410963072962
