L(s) = 1 | − i·2-s − 4-s + (1 − 2i)5-s + i·7-s + i·8-s + (−2 − i)10-s + 3·11-s + 4i·13-s + 14-s + 16-s + 19-s + (−1 + 2i)20-s − 3i·22-s + 5i·23-s + (−3 − 4i)25-s + 4·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.377i·7-s + 0.353i·8-s + (−0.632 − 0.316i)10-s + 0.904·11-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s + 0.229·19-s + (−0.223 + 0.447i)20-s − 0.639i·22-s + 1.04i·23-s + (−0.600 − 0.800i)25-s + 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.935445104\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.935445104\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 5iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.108393999182225739941639648002, −8.253766508185137760681900412881, −7.31056034392922677186730411044, −6.24903876178345709772211330298, −5.63780248374158553890614914115, −4.59690364956391631141625247499, −4.12131856928668054571521631682, −2.94097609371198186238399615557, −1.83752768207418797841057354295, −1.11379638749635992559441267076,
0.74370061332604406159554930944, 2.25213487854780927025181149392, 3.33136440373610905353862539951, 4.09133533823063571306132515451, 5.17423726607114961308056746940, 5.97114252062701652524117938244, 6.57578117983521645876269268085, 7.28234950793085691563411345893, 7.910757319965529520357772887768, 8.854691586096177477869225921079