| L(s) = 1 | − i·2-s − 4-s − 4i·7-s + i·8-s + i·13-s − 4·14-s + 16-s − 3i·17-s + 4·19-s + 26-s + 4i·28-s − 9·29-s − 4·31-s − i·32-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.51i·7-s + 0.353i·8-s + 0.277i·13-s − 1.06·14-s + 0.250·16-s − 0.727i·17-s + 0.917·19-s + 0.196·26-s + 0.755i·28-s − 1.67·29-s − 0.718·31-s − 0.176i·32-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9512736748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9512736748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−7.81900891248323063813513522436, −7.43440074110954368404356386468, −6.73582501922805070921745560488, −5.57894157718129079984176208177, −4.89942545675181924895369893307, −3.89034624218382587703627909221, −3.57067866155828413075964481022, −2.34938286126865355307774196393, −1.29328879913571905920769689735, −0.28430474715581416215483635426,
1.48858910180921083050740063540, 2.60607115193411289020490299936, 3.49041304798277421477860532803, 4.50304757635060855614619496632, 5.49799113904488327787955516363, 5.74879838736197856199114861138, 6.53613904393485464906030937297, 7.56364463833830175531353236062, 7.993936650441777623983882154210, 8.900544546637272776672958934026
