| L(s) = 1 | − 2.70·3-s + (−1.60 − 1.60i)7-s + 4.33·9-s + (3.90 − 3.90i)11-s − 1.48i·13-s + (4.26 + 4.26i)17-s + (−0.162 + 0.162i)19-s + (4.35 + 4.35i)21-s + (−5.87 + 5.87i)23-s − 3.60·27-s + (−1.48 − 1.48i)29-s + 6.78i·31-s + (−10.5 + 10.5i)33-s − 6.13i·37-s + 4.02i·39-s + ⋯ |
| L(s) = 1 | − 1.56·3-s + (−0.608 − 0.608i)7-s + 1.44·9-s + (1.17 − 1.17i)11-s − 0.412i·13-s + (1.03 + 1.03i)17-s + (−0.0373 + 0.0373i)19-s + (0.950 + 0.950i)21-s + (−1.22 + 1.22i)23-s − 0.694·27-s + (−0.276 − 0.276i)29-s + 1.21i·31-s + (−1.84 + 1.84i)33-s − 1.00i·37-s + 0.644i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6775534166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6775534166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 7 | \( 1 + (1.60 + 1.60i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.90 + 3.90i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 17 | \( 1 + (-4.26 - 4.26i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.162 - 0.162i)T - 19iT^{2} \) |
| 23 | \( 1 + (5.87 - 5.87i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.48 + 1.48i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.78iT - 31T^{2} \) |
| 37 | \( 1 + 6.13iT - 37T^{2} \) |
| 41 | \( 1 + 2.75iT - 41T^{2} \) |
| 43 | \( 1 + 3.39iT - 43T^{2} \) |
| 47 | \( 1 + (-9.44 + 9.44i)T - 47iT^{2} \) |
| 53 | \( 1 + 1.54T + 53T^{2} \) |
| 59 | \( 1 + (-2.53 - 2.53i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.600 + 0.600i)T - 61iT^{2} \) |
| 67 | \( 1 + 8.14iT - 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + (4.84 + 4.84i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.455T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 + (1.73 + 1.73i)T + 97iT^{2} \) |
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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
−9.344971826299716356824636596275, −8.342809350696958227191694749176, −7.32743939769193170385295233159, −6.55184967109098546939475864200, −5.81495944199351606751565961855, −5.45231761986496961553011720656, −3.95017090150685822074022129532, −3.55148217097614525194734095354, −1.46601571908552330046608925437, −0.39993540235504758683934866334,
1.12946547699848934318781206788, 2.52938821087773105831028759958, 4.06079202851724113366148183718, 4.72969548040088449937467409906, 5.71728305255391138992898873366, 6.34689803611125620188300167373, 6.91683343880442731418795433700, 7.86363372930826456754598866258, 9.204982502934874836182954907169, 9.699518436489407421747882843114
