| L(s) = 1 | − 2i·2-s + i·3-s − 4·4-s + 2·6-s + 18i·7-s + 8i·8-s + 26·9-s − 32·11-s − 4i·12-s − 47i·13-s + 36·14-s + 16·16-s − 20i·17-s − 52i·18-s − 36·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.192i·3-s − 0.5·4-s + 0.136·6-s + 0.971i·7-s + 0.353i·8-s + 0.962·9-s − 0.877·11-s − 0.0962i·12-s − 1.00i·13-s + 0.687·14-s + 0.250·16-s − 0.285i·17-s − 0.680i·18-s − 0.434·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.754656450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754656450\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23iT \) |
| good | 3 | \( 1 - iT - 27T^{2} \) |
| 7 | \( 1 - 18iT - 343T^{2} \) |
| 11 | \( 1 + 32T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 20iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 36T + 6.85e3T^{2} \) |
| 29 | \( 1 - 27T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 157T + 6.89e4T^{2} \) |
| 43 | \( 1 - 18iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 65iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 14iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 744T + 2.05e5T^{2} \) |
| 61 | \( 1 - 552T + 2.26e5T^{2} \) |
| 67 | \( 1 - 156iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 699T + 3.57e5T^{2} \) |
| 73 | \( 1 + 609iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 644T + 4.93e5T^{2} \) |
| 83 | \( 1 - 512iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 102T + 7.04e5T^{2} \) |
| 97 | \( 1 + 578iT - 9.12e5T^{2} \) |
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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
−9.453268268429156239021701226952, −8.546120481622535361352528206214, −7.86789926941610879084074293664, −6.79279338172748989328886110233, −5.56384174029250106696254186452, −5.01239495065470418242703921624, −3.89754864107416668987693964554, −2.83914000070068142134147479566, −2.00473943284263327042443923987, −0.57110409237879660682512388797,
0.827849945342552178613483849407, 2.07391936084045200963661703409, 3.70700424576106211360027109790, 4.40771840936118259395765769420, 5.30070672834448934746670885091, 6.52553877139369898041758517807, 7.02980907065425941997839378711, 7.76759934325251520322990066354, 8.545185064752985026744878051610, 9.611623315354084441004582073718
