| L(s) = 1 | + (1.40 + 0.132i)2-s + (0.453 + 0.891i)3-s + (1.96 + 0.374i)4-s + (−0.600 − 2.15i)5-s + (0.520 + 1.31i)6-s + (0.972 + 0.972i)7-s + (2.71 + 0.788i)8-s + (−0.587 + 0.809i)9-s + (−0.559 − 3.11i)10-s + (0.162 + 0.223i)11-s + (0.558 + 1.92i)12-s + (0.0801 + 0.505i)13-s + (1.23 + 1.49i)14-s + (1.64 − 1.51i)15-s + (3.71 + 1.47i)16-s + (−4.63 − 2.36i)17-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.0939i)2-s + (0.262 + 0.514i)3-s + (0.982 + 0.187i)4-s + (−0.268 − 0.963i)5-s + (0.212 + 0.536i)6-s + (0.367 + 0.367i)7-s + (0.960 + 0.278i)8-s + (−0.195 + 0.269i)9-s + (−0.176 − 0.984i)10-s + (0.0488 + 0.0672i)11-s + (0.161 + 0.554i)12-s + (0.0222 + 0.140i)13-s + (0.331 + 0.400i)14-s + (0.425 − 0.390i)15-s + (0.929 + 0.367i)16-s + (−1.12 − 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.41654 + 0.395372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.41654 + 0.395372i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.132i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (0.600 + 2.15i)T \) |
| good | 7 | \( 1 + (-0.972 - 0.972i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.162 - 0.223i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0801 - 0.505i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (4.63 + 2.36i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.118 - 0.363i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.0164 - 0.104i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (5.13 + 1.66i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.03 + 1.31i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (9.32 - 1.47i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.259 + 0.188i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (6.23 - 6.23i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.85 + 1.45i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-8.32 + 4.24i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (11.0 + 8.03i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.38 + 6.81i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.33 - 2.61i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-11.5 - 3.74i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.09 - 0.490i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (4.72 - 14.5i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (12.6 + 6.42i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (2.29 + 3.16i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.80 - 11.3i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.76773022681424127492698934437, −11.26724524393893822163163227351, −9.948822438840798620588415301506, −8.827842863508856776123566560596, −8.014096660684226623752517802584, −6.76091142847509882825503915748, −5.39369534708407138008135537568, −4.70470129588140185112664903140, −3.68253027718394922609982155890, −2.09131464106628464055566692301,
1.98614398730522384883388148898, 3.26015631234316806186604507737, 4.32939813339517827858320771195, 5.80015913798945608283130373225, 6.83771951072532650530120884187, 7.43523292719886341148333661492, 8.625033205869771449131207944688, 10.27830117882193269810627493517, 10.95057844327073515309301091053, 11.75552774879386175441608151118
