L(s) = 1 | + (2.78 − 0.501i)2-s + (7.49 − 2.79i)4-s + 4.57i·5-s + (−2.93 − 18.2i)7-s + (19.4 − 11.5i)8-s + (2.29 + 12.7i)10-s − 26.0i·11-s − 75.0i·13-s + (−17.3 − 49.4i)14-s + (48.4 − 41.8i)16-s + 115. i·17-s + 119.·19-s + (12.7 + 34.3i)20-s + (−13.0 − 72.4i)22-s − 61.4i·23-s + ⋯ |
L(s) = 1 | + (0.984 − 0.177i)2-s + (0.937 − 0.348i)4-s + 0.409i·5-s + (−0.158 − 0.987i)7-s + (0.860 − 0.509i)8-s + (0.0725 + 0.402i)10-s − 0.713i·11-s − 1.60i·13-s + (−0.330 − 0.943i)14-s + (0.756 − 0.654i)16-s + 1.65i·17-s + 1.43·19-s + (0.142 + 0.383i)20-s + (−0.126 − 0.701i)22-s − 0.556i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.368315556\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.368315556\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.78 + 0.501i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.93 + 18.2i)T \) |
good | 5 | \( 1 - 4.57iT - 125T^{2} \) |
| 11 | \( 1 + 26.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 75.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 115. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 71.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 144. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 288. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 403.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 21.7iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 598. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 589. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 6.85iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 972. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 429.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3iT - 9.12e5T^{2} \) |
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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
−11.34783970696562806571059764114, −10.63251408212367809221602114922, −10.04439773259244162420963685265, −8.227276820840638408486380562617, −7.28237680490630261281198047008, −6.21346623209582606795481894192, −5.24564270942528477479286488303, −3.78363114998550111499775013707, −3.01327119371387730133722718769, −1.05507404476045346342373935930,
1.85461091448879896791086251068, 3.16745444676721680116023060105, 4.69240356892935975233159446137, 5.35543298041721641020600899665, 6.69608421029830065687391619906, 7.47631341968318743161709736960, 8.970209597274217497332187537650, 9.673261381916724369806003901297, 11.37939302543335336906616798764, 11.84636313241915028864876560575