| L(s) = 1 | + 3i·3-s + (2 − i)5-s + i·7-s − 6·9-s + 3·11-s − i·13-s + (3 + 6i)15-s + 5i·17-s + 8·19-s − 3·21-s − 2i·23-s + (3 − 4i)25-s − 9i·27-s − 29-s + 2·31-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s − 2·9-s + 0.904·11-s − 0.277i·13-s + (0.774 + 1.54i)15-s + 1.21i·17-s + 1.83·19-s − 0.654·21-s − 0.417i·23-s + (0.600 − 0.800i)25-s − 1.73i·27-s − 0.185·29-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.214331432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214331432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 11iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 3iT - 97T^{2} \) |
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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
−9.444937754881889020976340612451, −8.743799939616966892473971278641, −8.174808188300188667863368179361, −6.69488843095879628863724483095, −5.83813556402404487514444157361, −5.27233249759591380841527085999, −4.53032506320539930054225881121, −3.64433858944728584870714096975, −2.81820823696117805048741695195, −1.36931380178288279734317856982,
0.850475093751458666498531711327, 1.68738533447412663463829885713, 2.63251548610733233969776936747, 3.55109348064170407627398046915, 5.14621923235676818267446416094, 5.82681068969927050468553979772, 6.68939003075776498415699966681, 7.16800269198948105149103308177, 7.63401815350136425103219077516, 8.806347243927272990309576059979
