| 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
−12.68488374328286023646959267845, −12.17440655712224376192950634342, −11.12296729796376880015420834771, −9.780175415406131581848214914772, −8.158433267743796917087079486042, −7.26240524138543197046228235259, −6.70070602622316409333741506782, −5.24717738160555798387510735840, −2.65246387966730427807742787224, −1.80755974444461894317313577560,
2.86410613473193407254618297336, 4.42760603249878175042577894921, 5.63174381352013915887914532852, 6.30316325164218237662425962233, 8.324311384890622410343799030693, 9.388338275339866410748577063799, 10.35682777937643509638023719713, 10.90419090966603120975729203665, 11.76614926898526895368768527787, 13.85749078200013579585050329943
