L(s) = 1 | + (1.53 + 1.11i)2-s + (0.363 − 1.11i)3-s + (0.809 + 2.48i)4-s + (−0.809 + 0.587i)5-s + (1.80 − 1.31i)6-s + (−0.951 + 2.92i)8-s + (−0.309 − 0.224i)9-s − 1.90·10-s + (0.809 + 0.587i)11-s + 3.07·12-s + (−1.53 − 1.11i)13-s + (0.363 + 1.11i)15-s + (−2.61 + 1.90i)16-s + (−0.224 − 0.690i)18-s + (0.309 − 0.951i)19-s + (−2.11 − 1.53i)20-s + ⋯ |
L(s) = 1 | + (1.53 + 1.11i)2-s + (0.363 − 1.11i)3-s + (0.809 + 2.48i)4-s + (−0.809 + 0.587i)5-s + (1.80 − 1.31i)6-s + (−0.951 + 2.92i)8-s + (−0.309 − 0.224i)9-s − 1.90·10-s + (0.809 + 0.587i)11-s + 3.07·12-s + (−1.53 − 1.11i)13-s + (0.363 + 1.11i)15-s + (−2.61 + 1.90i)16-s + (−0.224 − 0.690i)18-s + (0.309 − 0.951i)19-s + (−2.11 − 1.53i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.257373265\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.257373265\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + 1.17T + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.51207138320192999849417852626, −9.042955851899801053762586038777, −7.85047310982430005674454660748, −7.52737249022581230994280281071, −6.99790038976445101802331321535, −6.28284426519682486093475468628, −5.08558100409130237442060242380, −4.31805634857752440322283090780, −3.15883124590015138532929464011, −2.42039324512963751380082861884,
1.63417776765186976561676001238, 3.13035201726454912294421340304, 3.75297522989244152125670996512, 4.57676258766415531990716196898, 4.92532442785225242233063850471, 6.16098803102305698148127688976, 7.23630491207833766671716994596, 8.688378028161994001175715408097, 9.554170882112875940474216188239, 10.00734203391120836527009224745