L(s) = 1 | + i·2-s − 0.519i·3-s − 4-s + 0.519·6-s + 4.76i·7-s − i·8-s + 2.72·9-s + 0.960·11-s + 0.519i·12-s − 2.24i·13-s − 4.76·14-s + 16-s − 0.249i·17-s + 2.72i·18-s − 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.300i·3-s − 0.5·4-s + 0.212·6-s + 1.80i·7-s − 0.353i·8-s + 0.909·9-s + 0.289·11-s + 0.150i·12-s − 0.623i·13-s − 1.27·14-s + 0.250·16-s − 0.0605i·17-s + 0.643i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.762390 + 1.23357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.762390 + 1.23357i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 0.519iT - 3T^{2} \) |
| 7 | \( 1 - 4.76iT - 7T^{2} \) |
| 11 | \( 1 - 0.960T + 11T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 0.249iT - 17T^{2} \) |
| 23 | \( 1 - 9.01iT - 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 0.0399iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.96iT - 43T^{2} \) |
| 47 | \( 1 - 9.49iT - 47T^{2} \) |
| 53 | \( 1 - 6.84iT - 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 5.72iT - 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 7.45iT - 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.930561436917210977261436237447, −9.391439982187568834829145488967, −8.564054508914421733330955289655, −7.75881970865475145499142657471, −6.95428267861157701658451155075, −5.86413257852788367182680816471, −5.46567926164392620434802941542, −4.23765820805346630761349210753, −2.93137557232701202077408354688, −1.59737522439635978392269332245,
0.72714030816066104783210256445, 2.01469847215699411061785591196, 3.70441951959950249883393506760, 4.13547085172331797934534427530, 4.96322753448470782834708735263, 6.60094737047827828878484941912, 7.13525243481190563249699701466, 8.187947357118248153565396267152, 9.172606579244305475733623021261, 10.14305831671278914549861073295