L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3.38·7-s − 8-s + 9-s − 6.27·11-s + 12-s − 6.89·13-s + 3.38·14-s + 16-s − 1.40·17-s − 18-s + 1.97·19-s − 3.38·21-s + 6.27·22-s + 2.40·23-s − 24-s + 6.89·26-s + 27-s − 3.38·28-s + 7.79·29-s + 31-s − 32-s − 6.27·33-s + 1.40·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s − 1.89·11-s + 0.288·12-s − 1.91·13-s + 0.903·14-s + 0.250·16-s − 0.341·17-s − 0.235·18-s + 0.452·19-s − 0.737·21-s + 1.33·22-s + 0.502·23-s − 0.204·24-s + 1.35·26-s + 0.192·27-s − 0.639·28-s + 1.44·29-s + 0.179·31-s − 0.176·32-s − 1.09·33-s + 0.241·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6620027787\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6620027787\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 6.27T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 1.97T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 2.59T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 + 0.973T + 59T^{2} \) |
| 61 | \( 1 - 7.40T + 61T^{2} \) |
| 67 | \( 1 - 1.92T + 67T^{2} \) |
| 71 | \( 1 - 0.540T + 71T^{2} \) |
| 73 | \( 1 - 1.43T + 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.212233881485141908455078435313, −7.71360560779416468981820138453, −7.00552193269786885823576188805, −6.46306352562849734360313829422, −5.25608514500477468068167559162, −4.77772370887691440953030217123, −3.31042920044614455760690632575, −2.78672028627843652945910288892, −2.17514990673592332232179133642, −0.44867687219676175154864140219,
0.44867687219676175154864140219, 2.17514990673592332232179133642, 2.78672028627843652945910288892, 3.31042920044614455760690632575, 4.77772370887691440953030217123, 5.25608514500477468068167559162, 6.46306352562849734360313829422, 7.00552193269786885823576188805, 7.71360560779416468981820138453, 8.212233881485141908455078435313