| L(s) = 1 | + i·2-s + i·3-s − 4-s + (−1 − 2i)5-s − 6-s − i·8-s − 9-s + (2 − i)10-s − i·12-s − 2i·13-s + (2 − i)15-s + 16-s + 2i·17-s − i·18-s − 2·19-s + (1 + 2i)20-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s − 0.288i·12-s − 0.554i·13-s + (0.516 − 0.258i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 0.458·19-s + (0.223 + 0.447i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171332500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171332500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.572708749936396748563503679621, −8.890633785899576545509713612509, −8.111934693486315210027938442103, −7.61730877381858834168326845308, −6.35883997049681781481436172051, −5.60888267973068842257700672129, −4.74267242887605239229249925805, −4.11167904436867214751675664336, −3.02216750350781253582513587081, −1.13406359887047313608195671437,
0.55256822690500779588170780157, 2.19072490866271890584532810028, 2.85169545012310337786235769235, 3.99387810715496440634896354350, 4.82753968056758351523614312838, 6.22773510895407485795309833849, 6.73873197140641973570619042430, 7.70505603548320458921216221758, 8.442496462794057327562558054761, 9.264703579205492149699191454553
