L(s) = 1 | + 1.10·5-s + (−2.64 − 0.0763i)7-s − 3.42i·11-s − 4.98i·13-s − 6.38·17-s + 4.65i·19-s + 8.98i·23-s − 3.77·25-s − 1.51i·29-s + 6.71i·31-s + (−2.92 − 0.0843i)35-s − 2.83·37-s − 10.1·41-s + 10.8·43-s − 12.8·47-s + ⋯ |
L(s) = 1 | + 0.494·5-s + (−0.999 − 0.0288i)7-s − 1.03i·11-s − 1.38i·13-s − 1.54·17-s + 1.06i·19-s + 1.87i·23-s − 0.755·25-s − 0.280i·29-s + 1.20i·31-s + (−0.494 − 0.0142i)35-s − 0.465·37-s − 1.57·41-s + 1.66·43-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1395607899\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1395607899\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0763i)T \) |
good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 + 3.42iT - 11T^{2} \) |
| 13 | \( 1 + 4.98iT - 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 - 4.65iT - 19T^{2} \) |
| 23 | \( 1 - 8.98iT - 23T^{2} \) |
| 29 | \( 1 + 1.51iT - 29T^{2} \) |
| 31 | \( 1 - 6.71iT - 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.02iT - 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 9.16iT - 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 + 8.94iT - 71T^{2} \) |
| 73 | \( 1 + 7.79iT - 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 + 5.73T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 8.04iT - 97T^{2} \) |
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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
−9.147272647841733533449356034090, −8.316290916184254043476494260202, −7.50529240297954039554189422171, −6.40558409396280784129891889894, −5.90702966154333490846555732134, −5.08895838208941644937349201750, −3.61131479779956903647859222305, −3.12260261885209879617638782513, −1.71201607705498305854253225766, −0.05015364249702227532163094132,
1.96514849567579159497331213560, 2.66784003577563732470093627733, 4.19863282061014550348328802819, 4.64281770188965859490144119943, 6.01601376084783209822436052776, 6.74221252882532413769191569260, 7.09201773188380534798435003536, 8.530458484024939120422288150933, 9.207636251068585250899016678634, 9.705945986603802840939035250895