| L(s) = 1 | − 2-s + i·3-s + 4-s − i·6-s + 2·7-s − 8-s − 9-s + i·12-s + (3 + 2i)13-s − 2·14-s + 16-s + 2i·17-s + 18-s − i·19-s + 2i·21-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.408i·6-s + 0.755·7-s − 0.353·8-s − 0.333·9-s + 0.288i·12-s + (0.832 + 0.554i)13-s − 0.534·14-s + 0.250·16-s + 0.485i·17-s + 0.235·18-s − 0.229i·19-s + 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300541978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300541978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.289652542310655931348926483791, −8.699895026832717351423229167487, −7.954416493368725825981979816299, −7.21575761657168747962306120766, −6.16221352636380788981600596621, −5.46041752957976620860636171966, −4.36723014899708923145539651733, −3.59162767496938085843594843250, −2.30369716259958318794779535811, −1.21602070870328989511270092443,
0.67403979531714114889609173213, 1.75940078012088225073514153953, 2.76599574179595186235220927712, 3.94055344507768358818709446771, 5.15318263255546329775953306721, 5.96577361513691795225601571884, 6.79609357676426913739945188046, 7.62037735314479048695969179586, 8.234306397243358122292247304650, 8.786149615449507024000656424940
