| L(s) = 1 | + (0.998 + 0.0523i)2-s + (−0.658 − 0.813i)3-s + (0.994 + 0.104i)4-s + (−0.0376 + 2.23i)5-s + (−0.615 − 0.846i)6-s + (−0.124 − 2.64i)7-s + (0.987 + 0.156i)8-s + (0.395 − 1.86i)9-s + (−0.154 + 2.23i)10-s + (3.95 − 0.841i)11-s + (−0.570 − 0.877i)12-s + (3.93 + 2.00i)13-s + (0.0136 − 2.64i)14-s + (1.84 − 1.44i)15-s + (0.978 + 0.207i)16-s + (−0.193 − 0.503i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.0370i)2-s + (−0.380 − 0.469i)3-s + (0.497 + 0.0522i)4-s + (−0.0168 + 0.999i)5-s + (−0.251 − 0.345i)6-s + (−0.0471 − 0.998i)7-s + (0.349 + 0.0553i)8-s + (0.131 − 0.620i)9-s + (−0.0488 + 0.705i)10-s + (1.19 − 0.253i)11-s + (−0.164 − 0.253i)12-s + (1.09 + 0.556i)13-s + (0.00364 − 0.707i)14-s + (0.476 − 0.372i)15-s + (0.244 + 0.0519i)16-s + (−0.0468 − 0.122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88664 - 0.317154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88664 - 0.317154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 0.0523i)T \) |
| 5 | \( 1 + (0.0376 - 2.23i)T \) |
| 7 | \( 1 + (0.124 + 2.64i)T \) |
| good | 3 | \( 1 + (0.658 + 0.813i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-3.95 + 0.841i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.93 - 2.00i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.193 + 0.503i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.577 - 5.49i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.317 + 6.04i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (3.41 - 4.69i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.85 + 4.17i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (9.82 - 6.37i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-5.21 + 1.69i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.62 + 5.62i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.20 + 1.61i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-5.29 + 4.28i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (4.81 - 5.34i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (7.57 - 6.81i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.93 + 1.51i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-11.9 - 8.71i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.01 - 9.26i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-4.65 - 10.4i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.0505 - 0.319i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (12.0 + 13.4i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.275 - 1.74i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.54299950376319653929491905959, −10.78678230570004962579002420539, −9.873792782693435931931479384176, −8.488680706061856585328926792028, −7.06839493666506537510131863038, −6.67424693263685336930032174237, −5.89138704592318185585370920543, −4.03767209772677435364742101710, −3.50447631238939018423327737367, −1.47358363170025575268942895911,
1.73001938648464702069952393769, 3.54283335284083572726626145249, 4.70270451063143720569275065052, 5.44410666134917371952067449913, 6.32589037030523008639835790604, 7.79702378118083116019254587568, 8.911977699588545769223424695920, 9.583195068448011865168932521482, 10.96438415622516688179931087188, 11.58602056690270112932355886617
