L(s) = 1 | + (0.647 − 1.25i)2-s + (0.707 − 0.707i)3-s + (−1.16 − 1.62i)4-s + (2.52 + 2.52i)5-s + (−0.430 − 1.34i)6-s − i·7-s + (−2.79 + 0.403i)8-s − 1.00i·9-s + (4.80 − 1.53i)10-s + (−1.92 − 1.92i)11-s + (−1.97 − 0.331i)12-s + (4.58 − 4.58i)13-s + (−1.25 − 0.647i)14-s + 3.56·15-s + (−1.30 + 3.78i)16-s + 3.46·17-s + ⋯ |
L(s) = 1 | + (0.458 − 0.888i)2-s + (0.408 − 0.408i)3-s + (−0.580 − 0.814i)4-s + (1.12 + 1.12i)5-s + (−0.175 − 0.549i)6-s − 0.377i·7-s + (−0.989 + 0.142i)8-s − 0.333i·9-s + (1.51 − 0.485i)10-s + (−0.580 − 0.580i)11-s + (−0.569 − 0.0956i)12-s + (1.27 − 1.27i)13-s + (−0.335 − 0.173i)14-s + 0.920·15-s + (−0.326 + 0.945i)16-s + 0.840·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48252 - 1.39620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48252 - 1.39620i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.647 + 1.25i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + (-2.52 - 2.52i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.92 + 1.92i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.58 + 4.58i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + (3.90 - 3.90i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.95iT - 23T^{2} \) |
| 29 | \( 1 + (1.67 - 1.67i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 + (3.39 + 3.39i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.22iT - 41T^{2} \) |
| 43 | \( 1 + (-3.36 - 3.36i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 + (-0.328 - 0.328i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.92 - 4.92i)T + 59iT^{2} \) |
| 61 | \( 1 + (10.0 - 10.0i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.72 - 5.72i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.86iT - 71T^{2} \) |
| 73 | \( 1 + 6.93iT - 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 + (-3.79 + 3.79i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.71iT - 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96478625237840439013902691682, −10.63547764371627607072651063616, −9.840907888015187987477384084482, −8.682696352130215926102002391782, −7.52268443294204338989320188272, −6.02083727926857049494399528560, −5.70320400324429594168577552036, −3.62493065001143499639921119311, −2.90033398798048310491375924282, −1.53377442871357088268151213795,
2.12908352369080207803620936917, 3.94095632991276000735864055264, 4.96132550246109443794636534997, 5.73886185691820447541061515296, 6.80069777880332423594771845259, 8.195076200613089492956695400983, 9.029312217843223549834620824423, 9.369056507150606774151163682765, 10.72761817145933475455781454866, 12.19847222463927076612341252167