L(s) = 1 | + (0.0703 − 0.0703i)3-s + (2.58 − 0.562i)7-s + 2.99i·9-s + 0.777·11-s + (3.93 − 3.93i)13-s + (−0.982 − 0.982i)17-s − 1.14·19-s + (0.142 − 0.221i)21-s + (1.46 + 1.46i)23-s + (0.421 + 0.421i)27-s + 4.69i·29-s − 6.45i·31-s + (0.0546 − 0.0546i)33-s + (1.30 − 1.30i)37-s − 0.553i·39-s + ⋯ |
L(s) = 1 | + (0.0405 − 0.0405i)3-s + (0.977 − 0.212i)7-s + 0.996i·9-s + 0.234·11-s + (1.09 − 1.09i)13-s + (−0.238 − 0.238i)17-s − 0.263·19-s + (0.0310 − 0.0482i)21-s + (0.304 + 0.304i)23-s + (0.0810 + 0.0810i)27-s + 0.870i·29-s − 1.15i·31-s + (0.00951 − 0.00951i)33-s + (0.214 − 0.214i)37-s − 0.0886i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.029830920\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.029830920\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.58 + 0.562i)T \) |
good | 3 | \( 1 + (-0.0703 + 0.0703i)T - 3iT^{2} \) |
| 11 | \( 1 - 0.777T + 11T^{2} \) |
| 13 | \( 1 + (-3.93 + 3.93i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.982 + 0.982i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + (-1.46 - 1.46i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.69iT - 29T^{2} \) |
| 31 | \( 1 + 6.45iT - 31T^{2} \) |
| 37 | \( 1 + (-1.30 + 1.30i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.81iT - 41T^{2} \) |
| 43 | \( 1 + (-7.13 - 7.13i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.34 + 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.08 - 2.08i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 5.88iT - 61T^{2} \) |
| 67 | \( 1 + (-6.30 + 6.30i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (7.23 - 7.23i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.83iT - 79T^{2} \) |
| 83 | \( 1 + (-10.4 + 10.4i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 + (8.50 + 8.50i)T + 97iT^{2} \) |
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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
−9.583164142496929373136210810057, −8.487179055306210350665760924307, −8.065770111199815858013217093637, −7.31778030721775806583831302406, −6.20276314232438561430539157517, −5.30951716829122263462725150732, −4.58526191901567218335499111603, −3.50166838839367520191052865347, −2.28007193903222604874151815823, −1.11593685342319647898426613216,
1.12285881331533927197122699009, 2.24850627452409251558829559683, 3.70735233990491745549623118359, 4.30905182283915505299166110855, 5.43611024278793245892554506089, 6.37168452157795137350115703054, 6.98113366724158378750189187328, 8.153794762331315034273488986154, 8.828787134381232266248602253887, 9.298649525767639753079430038088