| L(s) = 1 | + (0.143 + 0.442i)2-s + (−0.207 + 3.29i)3-s + (1.44 − 1.04i)4-s + (1.43 + 0.180i)5-s + (−1.48 + 0.381i)6-s + (0.177 − 0.378i)7-s + (1.42 + 1.03i)8-s + (−7.81 − 0.987i)9-s + (0.125 + 0.658i)10-s + (−0.407 − 6.47i)11-s + (3.15 + 4.96i)12-s + (−2.69 + 3.25i)13-s + (0.192 + 0.0243i)14-s + (−0.892 + 4.67i)15-s + (0.850 − 2.61i)16-s + (0.584 + 1.24i)17-s + ⋯ |
| L(s) = 1 | + (0.101 + 0.312i)2-s + (−0.119 + 1.90i)3-s + (0.721 − 0.524i)4-s + (0.640 + 0.0809i)5-s + (−0.606 + 0.155i)6-s + (0.0672 − 0.142i)7-s + (0.503 + 0.365i)8-s + (−2.60 − 0.329i)9-s + (0.0397 + 0.208i)10-s + (−0.122 − 1.95i)11-s + (0.910 + 1.43i)12-s + (−0.746 + 0.902i)13-s + (0.0514 + 0.00650i)14-s + (−0.230 + 1.20i)15-s + (0.212 − 0.654i)16-s + (0.141 + 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975441 + 0.938915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975441 + 0.938915i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-10.0 - 7.14i)T \) |
| good | 2 | \( 1 + (-0.143 - 0.442i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.207 - 3.29i)T + (-2.97 - 0.375i)T^{2} \) |
| 5 | \( 1 + (-1.43 - 0.180i)T + (4.84 + 1.24i)T^{2} \) |
| 7 | \( 1 + (-0.177 + 0.378i)T + (-4.46 - 5.39i)T^{2} \) |
| 11 | \( 1 + (0.407 + 6.47i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (2.69 - 3.25i)T + (-2.43 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.584 - 1.24i)T + (-10.8 + 13.0i)T^{2} \) |
| 19 | \( 1 + (3.04 - 2.21i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.70 + 2.69i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.32 + 1.31i)T + (21.1 - 19.8i)T^{2} \) |
| 31 | \( 1 + (-1.50 + 1.40i)T + (1.94 - 30.9i)T^{2} \) |
| 37 | \( 1 + (3.52 - 4.25i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (-6.83 + 1.75i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (0.138 + 0.294i)T + (-27.4 + 33.1i)T^{2} \) |
| 47 | \( 1 + (9.50 - 2.44i)T + (41.1 - 22.6i)T^{2} \) |
| 53 | \( 1 + (2.55 - 4.02i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (1.68 - 5.17i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.558 - 8.86i)T + (-60.5 + 7.64i)T^{2} \) |
| 67 | \( 1 + (-2.36 + 0.298i)T + (64.8 - 16.6i)T^{2} \) |
| 71 | \( 1 + (-4.48 + 9.52i)T + (-45.2 - 54.7i)T^{2} \) |
| 73 | \( 1 + (-0.237 + 0.505i)T + (-46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (2.47 + 2.32i)T + (4.96 + 78.8i)T^{2} \) |
| 83 | \( 1 + (6.59 + 3.62i)T + (44.4 + 70.0i)T^{2} \) |
| 89 | \( 1 + (5.66 + 3.11i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (9.57 - 1.20i)T + (93.9 - 24.1i)T^{2} \) |
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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
−13.85749078200013579585050329943, −11.76614926898526895368768527787, −10.90419090966603120975729203665, −10.35682777937643509638023719713, −9.388338275339866410748577063799, −8.324311384890622410343799030693, −6.30316325164218237662425962233, −5.63174381352013915887914532852, −4.42760603249878175042577894921, −2.86410613473193407254618297336,
1.80755974444461894317313577560, 2.65246387966730427807742787224, 5.24717738160555798387510735840, 6.70070602622316409333741506782, 7.26240524138543197046228235259, 8.158433267743796917087079486042, 9.780175415406131581848214914772, 11.12296729796376880015420834771, 12.17440655712224376192950634342, 12.68488374328286023646959267845
