L(s) = 1 | + (−1.17 + 0.788i)2-s − i·3-s + (0.755 − 1.85i)4-s − i·5-s + (0.788 + 1.17i)6-s − 0.567·7-s + (0.574 + 2.76i)8-s − 9-s + (0.788 + 1.17i)10-s + 2.37·11-s + (−1.85 − 0.755i)12-s − 2.37·13-s + (0.665 − 0.447i)14-s − 15-s + (−2.85 − 2.79i)16-s − 1.13i·17-s + ⋯ |
L(s) = 1 | + (−0.829 + 0.557i)2-s − 0.577i·3-s + (0.377 − 0.925i)4-s − 0.447i·5-s + (0.322 + 0.479i)6-s − 0.214·7-s + (0.203 + 0.979i)8-s − 0.333·9-s + (0.249 + 0.371i)10-s + 0.715·11-s + (−0.534 − 0.217i)12-s − 0.658·13-s + (0.177 − 0.119i)14-s − 0.258·15-s + (−0.714 − 0.699i)16-s − 0.276i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4140254764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4140254764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.788i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (3.57 + 3.19i)T \) |
good | 7 | \( 1 + 0.567T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 + 1.13iT - 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 8.93iT - 37T^{2} \) |
| 41 | \( 1 + 0.300T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 - 0.733iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 5.02iT - 59T^{2} \) |
| 61 | \( 1 - 9.36iT - 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 5.30iT - 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 - 7.56T + 83T^{2} \) |
| 89 | \( 1 - 14.2iT - 89T^{2} \) |
| 97 | \( 1 - 14.4iT - 97T^{2} \) |
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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
−9.204825388875348195290982826273, −8.265404576379218727785977845158, −7.81236070885356585273463715857, −6.70466725831292729752417655663, −6.31870348150127225193852157856, −5.25572329604139336610218263961, −4.27882157163263654839165713278, −2.63678036794890140758764721202, −1.54981314953682974048930719472, −0.21928233024167731425117240973,
1.64690930458240044932972502265, 2.86258727681650487304925261584, 3.68671446854721507562970136472, 4.59800061801225203148605342837, 5.96055678458001150077337883544, 6.84453269001363960807760653268, 7.60509014434382866826685268916, 8.596477993480130840320950806262, 9.207190614217766198942595576014, 9.999452594211018088051215096157