| L(s) = 1 | + (1.10 − 1.28i)2-s + (1.04 + 1.97i)3-s + (−0.130 − 0.867i)4-s + (−1.87 − 1.22i)5-s + (3.68 + 0.841i)6-s + (2.58 + 0.554i)7-s + (1.61 + 1.01i)8-s + (−1.10 + 1.62i)9-s + (−3.64 + 1.05i)10-s + (−0.325 + 0.222i)11-s + (1.57 − 1.16i)12-s + (0.523 + 1.49i)13-s + (3.57 − 2.71i)14-s + (0.459 − 4.96i)15-s + (4.76 − 1.46i)16-s + (−6.65 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (0.782 − 0.909i)2-s + (0.601 + 1.13i)3-s + (−0.0653 − 0.433i)4-s + (−0.837 − 0.546i)5-s + (1.50 + 0.343i)6-s + (0.977 + 0.209i)7-s + (0.570 + 0.358i)8-s + (−0.369 + 0.542i)9-s + (−1.15 + 0.333i)10-s + (−0.0981 + 0.0669i)11-s + (0.454 − 0.335i)12-s + (0.145 + 0.415i)13-s + (0.955 − 0.724i)14-s + (0.118 − 1.28i)15-s + (1.19 − 0.367i)16-s + (−1.61 − 0.704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09335 - 0.166392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09335 - 0.166392i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.87 + 1.22i)T \) |
| 7 | \( 1 + (-2.58 - 0.554i)T \) |
| good | 2 | \( 1 + (-1.10 + 1.28i)T + (-0.298 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-1.04 - 1.97i)T + (-1.68 + 2.47i)T^{2} \) |
| 11 | \( 1 + (0.325 - 0.222i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-0.523 - 1.49i)T + (-10.1 + 8.10i)T^{2} \) |
| 17 | \( 1 + (6.65 + 2.90i)T + (11.5 + 12.4i)T^{2} \) |
| 19 | \( 1 + (2.53 + 4.39i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.947 - 2.17i)T + (-15.6 + 16.8i)T^{2} \) |
| 29 | \( 1 + (4.74 + 3.78i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (3.05 + 1.76i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.881 - 1.19i)T + (-10.9 + 35.3i)T^{2} \) |
| 41 | \( 1 + (6.67 - 1.52i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (3.52 + 5.61i)T + (-18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (5.20 + 4.47i)T + (7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 1.89i)T + (-15.6 - 50.6i)T^{2} \) |
| 59 | \( 1 + (-9.59 - 8.90i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (2.22 - 14.7i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-1.56 + 0.419i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (4.90 + 6.14i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-5.96 + 5.13i)T + (10.8 - 72.1i)T^{2} \) |
| 79 | \( 1 + (-2.62 + 1.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.01 - 0.706i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (-1.50 - 1.02i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-12.9 + 12.9i)T - 97iT^{2} \) |
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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
−11.67076931425515879863063159036, −11.44318408014915436396184805085, −10.47245890811115530940947198027, −9.088397586256556616006178079509, −8.537986704411140885242465211272, −7.28531212363711992608480557397, −5.02134052002614531180485694229, −4.49290540805216697838847405358, −3.66988605842788081105303076890, −2.24344806232293168978932871821,
1.85719810287957600150667258984, 3.71678607414524220616582830123, 4.86860609760000499645012679704, 6.36782385775143103092877903197, 7.05854007361564538912859989425, 7.967616570103039344621680189312, 8.448452234568655845853753756827, 10.52016587580795961656445101179, 11.23311274869539732579644256980, 12.62417977270356381316074046336
