L(s) = 1 | − 1.99·2-s + 3-s + 1.97·4-s + 1.24·5-s − 1.99·6-s − 4.06·7-s + 0.0495·8-s + 9-s − 2.47·10-s + 3.04·11-s + 1.97·12-s + 2.14·13-s + 8.10·14-s + 1.24·15-s − 4.04·16-s + 17-s − 1.99·18-s + 2.62·19-s + 2.44·20-s − 4.06·21-s − 6.06·22-s + 0.657·23-s + 0.0495·24-s − 3.46·25-s − 4.27·26-s + 27-s − 8.02·28-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.577·3-s + 0.987·4-s + 0.554·5-s − 0.813·6-s − 1.53·7-s + 0.0175·8-s + 0.333·9-s − 0.781·10-s + 0.916·11-s + 0.570·12-s + 0.594·13-s + 2.16·14-s + 0.320·15-s − 1.01·16-s + 0.242·17-s − 0.469·18-s + 0.603·19-s + 0.547·20-s − 0.886·21-s − 1.29·22-s + 0.137·23-s + 0.0101·24-s − 0.692·25-s − 0.838·26-s + 0.192·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 - 3.04T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 - 0.657T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 9.42T + 31T^{2} \) |
| 37 | \( 1 - 0.281T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 0.718T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 7.51T + 67T^{2} \) |
| 71 | \( 1 - 4.69T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.285987800028965326923463860863, −7.43157044386731522234198231654, −6.82683788746232512044722674454, −6.21737107713128078443054581611, −5.28658356601524113483126995071, −3.80829160327354912018305773327, −3.38848375600911246708820300457, −2.12554711210968698813220049734, −1.35020668113318652333428413764, 0,
1.35020668113318652333428413764, 2.12554711210968698813220049734, 3.38848375600911246708820300457, 3.80829160327354912018305773327, 5.28658356601524113483126995071, 6.21737107713128078443054581611, 6.82683788746232512044722674454, 7.43157044386731522234198231654, 8.285987800028965326923463860863