| L(s) = 1 | + (−1.86 − 1.35i)2-s + (−2.32 − 1.03i)3-s + (1.02 + 3.16i)4-s + (1.24 − 2.16i)5-s + (2.93 + 5.08i)6-s + (1.07 − 1.19i)7-s + (0.944 − 2.90i)8-s + (2.31 + 2.57i)9-s + (−5.26 + 2.34i)10-s + (−0.717 − 0.152i)11-s + (0.883 − 8.40i)12-s + (0.198 + 1.88i)13-s + (−3.61 + 0.768i)14-s + (−5.13 + 3.73i)15-s + (−0.326 + 0.237i)16-s + (4.28 − 0.910i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 − 0.959i)2-s + (−1.34 − 0.597i)3-s + (0.513 + 1.58i)4-s + (0.558 − 0.967i)5-s + (1.19 + 2.07i)6-s + (0.405 − 0.449i)7-s + (0.333 − 1.02i)8-s + (0.772 + 0.858i)9-s + (−1.66 + 0.741i)10-s + (−0.216 − 0.0459i)11-s + (0.255 − 2.42i)12-s + (0.0549 + 0.522i)13-s + (−0.966 + 0.205i)14-s + (−1.32 + 0.964i)15-s + (−0.0817 + 0.0593i)16-s + (1.03 − 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133511 - 0.280085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133511 - 0.280085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-4.81 - 2.79i)T \) |
| good | 2 | \( 1 + (1.86 + 1.35i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (2.32 + 1.03i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 2.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.07 + 1.19i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (0.717 + 0.152i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.198 - 1.88i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-4.28 + 0.910i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.484 + 4.61i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.19 - 6.77i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.104 - 0.0757i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (4.21 + 7.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.73 - 2.99i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.0240 - 0.229i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-6.50 + 4.72i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.83 - 4.26i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-8.68 - 3.86i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.27 + 2.53i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (2.63 + 0.559i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (4.42 - 0.941i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.43 + 1.08i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.681 + 2.09i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.79 - 11.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−17.20239434336908949523521721763, −16.25757409502916935580679206252, −13.56161595118880043355780066316, −12.24896106010159871238148308966, −11.49677316992833898796817174529, −10.31712306705212862634028142470, −9.019462257302901746338478001628, −7.40334349263818985824558502958, −5.36466432647098909702482918333, −1.25116026881693198379894299837,
5.56940224862133110592340098817, 6.50447684464949805039729358218, 8.186334906577435185773998852218, 10.14875364090992357531118950640, 10.40691314721274135541632023182, 12.02175626228520112710779076517, 14.46482013116267318378340955297, 15.46937920600878073113721667185, 16.56348914007766398031830838452, 17.31993971380190430090177729115
