| L(s) = 1 | + (−0.988 − 0.149i)2-s + (1.88 − 1.28i)3-s + (0.955 + 0.294i)4-s + (0.269 + 3.59i)5-s + (−2.05 + 0.989i)6-s + (0.415 − 2.61i)7-s + (−0.900 − 0.433i)8-s + (0.804 − 2.05i)9-s + (0.269 − 3.59i)10-s + (−1.05 − 2.68i)11-s + (2.17 − 0.672i)12-s + (−1.54 + 1.94i)13-s + (−0.800 + 2.52i)14-s + (5.13 + 6.43i)15-s + (0.826 + 0.563i)16-s + (1.45 + 1.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (1.08 − 0.741i)3-s + (0.477 + 0.147i)4-s + (0.120 + 1.60i)5-s + (−0.839 + 0.404i)6-s + (0.157 − 0.987i)7-s + (−0.318 − 0.153i)8-s + (0.268 − 0.683i)9-s + (0.0852 − 1.13i)10-s + (−0.317 − 0.808i)11-s + (0.629 − 0.194i)12-s + (−0.429 + 0.538i)13-s + (−0.213 + 0.673i)14-s + (1.32 + 1.66i)15-s + (0.206 + 0.140i)16-s + (0.352 + 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.990675 - 0.146756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990675 - 0.146756i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (-0.415 + 2.61i)T \) |
| good | 3 | \( 1 + (-1.88 + 1.28i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.269 - 3.59i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.05 + 2.68i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (1.54 - 1.94i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.45 - 1.35i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.17 + 5.50i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.98 - 5.55i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.0991 + 0.434i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.567 - 0.982i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.67 + 1.44i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.93 - 0.930i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.92 - 1.41i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-10.0 - 1.50i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (11.2 + 3.48i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.383 + 5.12i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-14.0 + 4.34i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.137 + 0.238i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.34 + 5.90i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-7.92 + 1.19i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.51 - 4.35i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.53 + 8.19i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-0.966 + 2.46i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−14.03704357982247356912769628136, −13.14300496351529216350677062401, −11.39604935616899382491813166851, −10.64899502111515068339860778077, −9.534896153343059853557402919584, −8.085630572002323843811550057708, −7.37101086919205268499703675429, −6.46826876143160127199767111109, −3.48458011221641957399368276868, −2.25335281072874383160926498141,
2.27308574773387181212247625309, 4.37862443970960752812330060519, 5.69380805331391704561375036924, 8.018964568563883477196943648036, 8.513465617365720162994029750019, 9.486678998470133253385374089286, 10.14522244626028930681358948148, 12.13216947357894228999565439772, 12.67929946785654125468606651760, 14.29892758333637190830927534770
