| L(s) = 1 | + (−1.48 − 2.04i)2-s + (0.435 + 0.979i)3-s + (−1.35 + 4.16i)4-s + (−1.48 + 1.67i)5-s + (1.35 − 2.34i)6-s + (2.47 − 2.23i)7-s + (5.71 − 1.85i)8-s + (1.23 − 1.37i)9-s + (5.62 + 0.554i)10-s + (2.11 − 0.449i)11-s + (−4.66 + 0.490i)12-s + (5.65 + 0.594i)13-s + (−8.23 − 1.75i)14-s + (−2.28 − 0.726i)15-s + (−5.19 − 3.77i)16-s + (−1.26 + 5.96i)17-s + ⋯ |
| L(s) = 1 | + (−1.04 − 1.44i)2-s + (0.251 + 0.565i)3-s + (−0.676 + 2.08i)4-s + (−0.664 + 0.747i)5-s + (0.552 − 0.957i)6-s + (0.936 − 0.843i)7-s + (2.02 − 0.656i)8-s + (0.412 − 0.458i)9-s + (1.77 + 0.175i)10-s + (0.638 − 0.135i)11-s + (−1.34 + 0.141i)12-s + (1.56 + 0.164i)13-s + (−2.20 − 0.468i)14-s + (−0.589 − 0.187i)15-s + (−1.29 − 0.943i)16-s + (−0.307 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679342 - 0.318679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679342 - 0.318679i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.48 - 1.67i)T \) |
| 31 | \( 1 + (4.10 - 3.75i)T \) |
| good | 2 | \( 1 + (1.48 + 2.04i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.435 - 0.979i)T + (-2.00 + 2.22i)T^{2} \) |
| 7 | \( 1 + (-2.47 + 2.23i)T + (0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 0.449i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.65 - 0.594i)T + (12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.26 - 5.96i)T + (-15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0457 + 0.435i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (2.68 - 0.871i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.502 + 0.364i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1.47 + 0.850i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.45 + 2.42i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (3.37 - 0.354i)T + (42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-7.59 + 10.4i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.36 + 4.83i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-7.68 + 3.42i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 + (-5.07 + 2.93i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.31 - 2.56i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.55 - 7.33i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (3.33 + 0.708i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.461 + 1.03i)T + (-55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (5.07 - 15.6i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (16.5 + 5.38i)T + (78.4 + 57.0i)T^{2} \) |
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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
−12.35157759338862974347061995228, −11.33281985458954845495249153153, −10.78376304422752904352161132258, −10.12431215364432123498278553645, −8.777363823864926000650365822937, −8.170953660957561335934156508979, −6.78999145807676283581197269850, −3.88917357003851628999166427215, −3.75693776566795240989288633642, −1.50408774695761513715201240404,
1.39316643254480675273407936924, 4.63024182831461855759406722831, 5.78714704654079806646468994722, 7.08418208444886956782027279807, 8.013886387200611606228884697344, 8.586449231728405979105088836303, 9.386583888003629095205601831244, 11.00261299283135145317692232637, 12.04087754502166775539848408704, 13.41353642593321045605355001784
