| L(s) = 1 | + (−0.229 + 1.39i)2-s + (0.453 + 0.891i)3-s + (−1.89 − 0.640i)4-s + (−1.90 − 1.17i)5-s + (−1.34 + 0.429i)6-s + (−2.16 − 2.16i)7-s + (1.32 − 2.49i)8-s + (−0.587 + 0.809i)9-s + (2.07 − 2.38i)10-s + (−3.61 − 4.97i)11-s + (−0.289 − 1.97i)12-s + (0.749 + 4.72i)13-s + (3.51 − 2.52i)14-s + (0.182 − 2.22i)15-s + (3.18 + 2.42i)16-s + (−3.51 − 1.79i)17-s + ⋯ |
| L(s) = 1 | + (−0.162 + 0.986i)2-s + (0.262 + 0.514i)3-s + (−0.947 − 0.320i)4-s + (−0.850 − 0.525i)5-s + (−0.550 + 0.175i)6-s + (−0.817 − 0.817i)7-s + (0.469 − 0.882i)8-s + (−0.195 + 0.269i)9-s + (0.656 − 0.754i)10-s + (−1.09 − 1.50i)11-s + (−0.0836 − 0.571i)12-s + (0.207 + 1.31i)13-s + (0.939 − 0.674i)14-s + (0.0472 − 0.575i)15-s + (0.795 + 0.606i)16-s + (−0.852 − 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.212570 - 0.190908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212570 - 0.190908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.229 - 1.39i)T \) |
| 3 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (1.90 + 1.17i)T \) |
| good | 7 | \( 1 + (2.16 + 2.16i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.61 + 4.97i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.749 - 4.72i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.51 + 1.79i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.678 + 2.08i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.374 - 2.36i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.48 - 0.480i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (8.29 - 2.69i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 0.644i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (5.32 + 3.86i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.72 + 8.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.47 + 1.77i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (5.54 - 2.82i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (7.51 + 5.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.715 - 0.519i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.52 - 2.98i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (5.96 + 1.93i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.906 - 0.143i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (2.98 - 9.17i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.49 - 3.31i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.440 - 0.606i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (7.49 + 14.7i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.25894865087514272202587972523, −10.53753383962496492239222798306, −9.183777524891021562910950734306, −8.780235283789072352184387514582, −7.64915332523740916788479136550, −6.80283193019784757821217753089, −5.49560913066883826131174930649, −4.35180419026809052336466463706, −3.47841788587524706775579146090, −0.20671189636452171206374596557,
2.32379124843911931635455123032, 3.16020171876697180634666949226, 4.53943312211591746941084636884, 6.02455085239654330705462165300, 7.50205978359395073757541087010, 8.118587675130031674696667855249, 9.253962269589266136444500959059, 10.25713138488288071630735625494, 10.96848660765306129397666188434, 12.17722376719846933752573318912
