L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s + 1.19i·7-s − i·8-s + 9-s + 10-s + 0.554i·11-s + 12-s − 1.19·14-s + i·15-s + 16-s + 6.45·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.452i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.167i·11-s + 0.288·12-s − 0.320·14-s + 0.258i·15-s + 0.250·16-s + 1.56·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6180278370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6180278370\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.19iT - 7T^{2} \) |
| 11 | \( 1 - 0.554iT - 11T^{2} \) |
| 17 | \( 1 - 6.45T + 17T^{2} \) |
| 19 | \( 1 - 5.40iT - 19T^{2} \) |
| 23 | \( 1 + 7.96T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 3.04iT - 31T^{2} \) |
| 37 | \( 1 + 7.96iT - 37T^{2} \) |
| 41 | \( 1 + 3.08iT - 41T^{2} \) |
| 43 | \( 1 + 1.15T + 43T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 1.45iT - 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 3.34iT - 67T^{2} \) |
| 71 | \( 1 + 2.35iT - 71T^{2} \) |
| 73 | \( 1 - 13.3iT - 73T^{2} \) |
| 79 | \( 1 + 0.990T + 79T^{2} \) |
| 83 | \( 1 + 5.88iT - 83T^{2} \) |
| 89 | \( 1 - 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−8.410362078383009080676729765404, −7.73245149132397504086556934810, −7.34326522732372147769794377565, −6.13127913178980474025509439317, −5.71415665171712533297673324911, −5.33575833806180488770688304767, −4.13603556917027478751956895824, −3.72299465215407689294851324269, −2.23190292054828369381780175046, −1.16219775721449570606014574495,
0.20329314988200176488010316024, 1.31151364500920958810504746257, 2.35267125472025301286286419469, 3.39348862016188157831533201412, 3.93517234554547645527766715621, 4.94055317048889572877865981788, 5.57810955283070511859760050859, 6.37507468659551454099944413091, 7.19728591684086321345479555267, 7.81260942368793097641546606650