L(s) = 1 | + (0.976 + 1.02i)2-s + 0.502·3-s + (−0.0929 + 1.99i)4-s − 5-s + (0.490 + 0.513i)6-s + 3.57i·7-s + (−2.13 + 1.85i)8-s − 2.74·9-s + (−0.976 − 1.02i)10-s − 3.89i·11-s + (−0.0466 + 1.00i)12-s + 7.12i·13-s + (−3.65 + 3.49i)14-s − 0.502·15-s + (−3.98 − 0.371i)16-s + 6.79·17-s + ⋯ |
L(s) = 1 | + (0.690 + 0.723i)2-s + 0.289·3-s + (−0.0464 + 0.998i)4-s − 0.447·5-s + (0.200 + 0.209i)6-s + 1.35i·7-s + (−0.754 + 0.656i)8-s − 0.915·9-s + (−0.308 − 0.323i)10-s − 1.17i·11-s + (−0.0134 + 0.289i)12-s + 1.97i·13-s + (−0.978 + 0.933i)14-s − 0.129·15-s + (−0.995 − 0.0928i)16-s + 1.64·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.823272 + 1.55922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823272 + 1.55922i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.976 - 1.02i)T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (-3.48 + 2.62i)T \) |
good | 3 | \( 1 - 0.502T + 3T^{2} \) |
| 7 | \( 1 - 3.57iT - 7T^{2} \) |
| 11 | \( 1 + 3.89iT - 11T^{2} \) |
| 13 | \( 1 - 7.12iT - 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 23 | \( 1 + 3.22iT - 23T^{2} \) |
| 29 | \( 1 - 2.89iT - 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 - 1.99iT - 37T^{2} \) |
| 41 | \( 1 + 1.95iT - 41T^{2} \) |
| 43 | \( 1 + 1.00iT - 43T^{2} \) |
| 47 | \( 1 + 5.80iT - 47T^{2} \) |
| 53 | \( 1 + 1.75iT - 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 6.64T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.02iT - 83T^{2} \) |
| 89 | \( 1 - 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 9.60iT - 97T^{2} \) |
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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
−11.80959130958991847462512403908, −11.29869683706936653484929279445, −9.371740295758358554614770517746, −8.679825220115667025397882319889, −8.077760008661348125559059056610, −6.76366291366903619120819917948, −5.85197928461364931024810136000, −5.01829848122573876896510599517, −3.55441980281696234005740962076, −2.65005555886892710808692901013,
0.982235494661427704147272188070, 2.98181910992764691810412985378, 3.70340294355065242139666102330, 4.97433203915582142634201561618, 5.91756383913905884932692518203, 7.49253942264853063675883354131, 7.982708311949597874990537127200, 9.737847266267696822715245781024, 10.15543930693521464440678124833, 11.06151822396284955029103688434