L(s) = 1 | + (0.639 − 1.26i)2-s + (−0.730 + 1.57i)3-s + (−1.18 − 1.61i)4-s + (1.51 + 1.92i)6-s + 1.25·7-s + (−2.79 + 0.458i)8-s + (−1.93 − 2.29i)9-s − 3.02i·11-s + (3.39 − 0.677i)12-s + 5.65·13-s + (0.803 − 1.58i)14-s + (−1.20 + 3.81i)16-s + 2.45·17-s + (−4.12 + 0.972i)18-s − 1.77·19-s + ⋯ |
L(s) = 1 | + (0.452 − 0.891i)2-s + (−0.421 + 0.906i)3-s + (−0.590 − 0.806i)4-s + (0.618 + 0.786i)6-s + 0.474·7-s + (−0.986 + 0.162i)8-s + (−0.644 − 0.764i)9-s − 0.911i·11-s + (0.980 − 0.195i)12-s + 1.56·13-s + (0.214 − 0.423i)14-s + (−0.301 + 0.953i)16-s + 0.595·17-s + (−0.973 + 0.229i)18-s − 0.407·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18600 - 0.978375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18600 - 0.978375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.639 + 1.26i)T \) |
| 3 | \( 1 + (0.730 - 1.57i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.25T + 7T^{2} \) |
| 11 | \( 1 + 3.02iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 8.84iT - 23T^{2} \) |
| 29 | \( 1 - 3.79T + 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 + 7.57iT - 41T^{2} \) |
| 43 | \( 1 - 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 1.83iT - 47T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.91iT - 59T^{2} \) |
| 61 | \( 1 + 8.16iT - 61T^{2} \) |
| 67 | \( 1 - 8.50iT - 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 + 4.59iT - 73T^{2} \) |
| 79 | \( 1 - 7.36iT - 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.79878341156398041557064664947, −9.911827895222952611870773240826, −8.819951527467722823168681912805, −8.340209544721692784569053580177, −6.27159667925987160593433208929, −5.80170489671138059481524968524, −4.60337305609076720024814879770, −3.87615658031490423839204875572, −2.79502108516549502140358961063, −0.901196964274869653953794967521,
1.50536300195108651305721961199, 3.28020786826194499802406177542, 4.59905724093068899390410472141, 5.56653988843784429372460743043, 6.34140408328243255011969927505, 7.22015790105366818882787755418, 7.991440947330221822311803529229, 8.675329408761555135922746560254, 9.895225302974336416960572201683, 11.21307190827481726172873401304