| 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
−8.854691586096177477869225921079, −7.910757319965529520357772887768, −7.28234950793085691563411345893, −6.57578117983521645876269268085, −5.97114252062701652524117938244, −5.17423726607114961308056746940, −4.09133533823063571306132515451, −3.33136440373610905353862539951, −2.25213487854780927025181149392, −0.74370061332604406159554930944,
1.11379638749635992559441267076, 1.83752768207418797841057354295, 2.94097609371198186238399615557, 4.12131856928668054571521631682, 4.59690364956391631141625247499, 5.63780248374158553890614914115, 6.24903876178345709772211330298, 7.31056034392922677186730411044, 8.253766508185137760681900412881, 9.108393999182225739941639648002
