L(s) = 1 | + (−1 − i)2-s − 3-s + 2i·4-s + (−1 − 2i)5-s + (1 + i)6-s + (−1 − i)7-s + (2 − 2i)8-s + 9-s + (−1 + 3i)10-s + (−3 + 3i)11-s − 2i·12-s + 4i·13-s + 2i·14-s + (1 + 2i)15-s − 4·16-s + (−5 − 5i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s − 0.577·3-s + i·4-s + (−0.447 − 0.894i)5-s + (0.408 + 0.408i)6-s + (−0.377 − 0.377i)7-s + (0.707 − 0.707i)8-s + 0.333·9-s + (−0.316 + 0.948i)10-s + (−0.904 + 0.904i)11-s − 0.577i·12-s + 1.10i·13-s + 0.534i·14-s + (0.258 + 0.516i)15-s − 16-s + (−1.21 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + (5 + 5i)T + 17iT^{2} \) |
| 19 | \( 1 + (5 - 5i)T - 19iT^{2} \) |
| 23 | \( 1 + (1 - i)T - 23iT^{2} \) |
| 29 | \( 1 + (-5 - 5i)T + 29iT^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (5 - 5i)T - 47iT^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (5 + 5i)T + 59iT^{2} \) |
| 61 | \( 1 + (1 - i)T - 61iT^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + (-1 - i)T + 97iT^{2} \) |
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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
−11.46574759667380122410150261708, −10.59273309588687988988562107263, −9.625350653196612621442010844386, −8.787661221243612341942801285456, −7.62682221860671592295869042249, −6.72848495978425082631292628476, −4.85977465272441622298382063252, −4.02122800468219321565212115666, −2.00613862041050658725312290962, 0,
2.72972912538582356414716305014, 4.68328939645955649737669524800, 6.14015997276311683887837545920, 6.51326708974038905061816688266, 7.944923204248483363061287855152, 8.560454258715767800743701086765, 10.08753467867943356355994031591, 10.73284349028198014132831754272, 11.33281622379642241046405486843