| L(s) = 1 | + (−0.366 + 1.36i)2-s + (1.33 + 1.10i)3-s + (−2 + 1.41i)6-s + (2.38 − 1.15i)7-s + (−2 + 1.99i)8-s + (0.548 + 2.94i)9-s + (−3.67 − 2.12i)11-s + (3.53 + 3.53i)13-s + (0.707 + 3.67i)14-s + (−1.99 − 3.46i)16-s + (1.36 − 0.366i)17-s + (−4.22 − 0.330i)18-s + (−6.06 + 3.5i)19-s + (4.44 + 1.09i)21-s + (4.24 − 4.24i)22-s + (4.09 + 1.09i)23-s + ⋯ |
| L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.769 + 0.639i)3-s + (−0.816 + 0.577i)6-s + (0.899 − 0.436i)7-s + (−0.707 + 0.707i)8-s + (0.182 + 0.983i)9-s + (−1.10 − 0.639i)11-s + (0.980 + 0.980i)13-s + (0.188 + 0.981i)14-s + (−0.499 − 0.866i)16-s + (0.331 − 0.0887i)17-s + (−0.996 − 0.0779i)18-s + (−1.39 + 0.802i)19-s + (0.970 + 0.239i)21-s + (0.904 − 0.904i)22-s + (0.854 + 0.228i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.697593 + 1.65321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.697593 + 1.65321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.33 - 1.10i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.38 + 1.15i)T \) |
| good | 2 | \( 1 + (0.366 - 1.36i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.53 - 3.53i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.36 + 0.366i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.06 - 3.5i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.09 - 1.09i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.965 + 0.258i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (3.53 + 3.53i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.46 + 5.46i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.732 - 2.73i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.258 + 0.965i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-6.76 + 1.81i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 - 3.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1 + i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.77 + 13.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.07 + 7.07i)T - 97iT^{2} \) |
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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
−10.88026026944238142096807359347, −10.39997903214788163495189260545, −8.925854657952602628565868803399, −8.432876104389766698744305378657, −7.83074408489874648355620384831, −6.82638576680400663098652956309, −5.66954246835138511829402618154, −4.66423355066277632656087586523, −3.47533629721085880765523570330, −2.11071668443074213672614576032,
1.14042827167282646071970407426, 2.36569853500982379244478981913, 3.06294128445661924585292759233, 4.61396948030042215611991400713, 5.99368435427586904966254656538, 7.02086307616806451195252530378, 8.214348823615652421757163749006, 8.599549085403560052911887576896, 9.782950618971008868301119925563, 10.65234788886295667517097798237
