L(s) = 1 | + 2.56·2-s + 4.56·4-s + 5-s + 6.56·8-s + 2.56·10-s + 1.56·11-s − 0.438·13-s + 7.68·16-s − 0.438·17-s + 7.12·19-s + 4.56·20-s + 4·22-s − 3.12·23-s + 25-s − 1.12·26-s − 6.68·29-s + 6.56·32-s − 1.12·34-s + 6·37-s + 18.2·38-s + 6.56·40-s + 5.12·41-s + 0.876·43-s + 7.12·44-s − 8·46-s − 8.68·47-s + 2.56·50-s + ⋯ |
L(s) = 1 | + 1.81·2-s + 2.28·4-s + 0.447·5-s + 2.31·8-s + 0.810·10-s + 0.470·11-s − 0.121·13-s + 1.92·16-s − 0.106·17-s + 1.63·19-s + 1.01·20-s + 0.852·22-s − 0.651·23-s + 0.200·25-s − 0.220·26-s − 1.24·29-s + 1.15·32-s − 0.192·34-s + 0.986·37-s + 2.95·38-s + 1.03·40-s + 0.800·41-s + 0.133·43-s + 1.07·44-s − 1.17·46-s − 1.26·47-s + 0.362·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.210726298\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.210726298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 + 0.438T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 5.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.263081070238688992816501391228, −7.900134872272231254807302311517, −7.23650974347492718530831842047, −6.37921666328103650889584270599, −5.74426987712770206974630890643, −5.09174003181781604846780691314, −4.20653339449392641939367318517, −3.42298547931377773724439693850, −2.55120311309898115357314168439, −1.49133954237686356772030236148,
1.49133954237686356772030236148, 2.55120311309898115357314168439, 3.42298547931377773724439693850, 4.20653339449392641939367318517, 5.09174003181781604846780691314, 5.74426987712770206974630890643, 6.37921666328103650889584270599, 7.23650974347492718530831842047, 7.900134872272231254807302311517, 9.263081070238688992816501391228