L(s) = 1 | − 2-s + 2.16·3-s + 4-s + 2.86·5-s − 2.16·6-s − 4.99·7-s − 8-s + 1.68·9-s − 2.86·10-s − 4.98·11-s + 2.16·12-s + 5.51·13-s + 4.99·14-s + 6.20·15-s + 16-s + 3.85·17-s − 1.68·18-s − 1.97·19-s + 2.86·20-s − 10.8·21-s + 4.98·22-s − 1.30·23-s − 2.16·24-s + 3.21·25-s − 5.51·26-s − 2.85·27-s − 4.99·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.24·3-s + 0.5·4-s + 1.28·5-s − 0.883·6-s − 1.88·7-s − 0.353·8-s + 0.560·9-s − 0.906·10-s − 1.50·11-s + 0.624·12-s + 1.52·13-s + 1.33·14-s + 1.60·15-s + 0.250·16-s + 0.936·17-s − 0.396·18-s − 0.452·19-s + 0.640·20-s − 2.35·21-s + 1.06·22-s − 0.271·23-s − 0.441·24-s + 0.643·25-s − 1.08·26-s − 0.549·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 4.99T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 - 9.06T + 41T^{2} \) |
| 43 | \( 1 + 7.43T + 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 - 0.814T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 + 5.39T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 0.519T + 97T^{2} \) |
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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
−8.238118780469358628052555793775, −7.52426266281745871082244360269, −6.69470641139649582023325730475, −5.81963253818115212234265367341, −5.59415371684349476166725824567, −3.63397930471282798269953044616, −3.29056257838013052641153217000, −2.42668984595116289561073837303, −1.69002449566186484984083449113, 0,
1.69002449566186484984083449113, 2.42668984595116289561073837303, 3.29056257838013052641153217000, 3.63397930471282798269953044616, 5.59415371684349476166725824567, 5.81963253818115212234265367341, 6.69470641139649582023325730475, 7.52426266281745871082244360269, 8.238118780469358628052555793775