L(s) = 1 | + 1.50·2-s + 3-s + 0.272·4-s + 1.10·5-s + 1.50·6-s − 2.60·8-s + 9-s + 1.67·10-s + 3.74·11-s + 0.272·12-s + 4.04·13-s + 1.10·15-s − 4.47·16-s − 1.28·17-s + 1.50·18-s + 2.77·19-s + 0.302·20-s + 5.63·22-s + 8.14·23-s − 2.60·24-s − 3.77·25-s + 6.10·26-s + 27-s − 1.13·29-s + 1.67·30-s + 3.06·31-s − 1.53·32-s + ⋯ |
L(s) = 1 | + 1.06·2-s + 0.577·3-s + 0.136·4-s + 0.495·5-s + 0.615·6-s − 0.920·8-s + 0.333·9-s + 0.528·10-s + 1.12·11-s + 0.0786·12-s + 1.12·13-s + 0.286·15-s − 1.11·16-s − 0.311·17-s + 0.355·18-s + 0.636·19-s + 0.0675·20-s + 1.20·22-s + 1.69·23-s − 0.531·24-s − 0.754·25-s + 1.19·26-s + 0.192·27-s − 0.210·29-s + 0.305·30-s + 0.550·31-s − 0.270·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.960581829\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.960581829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 - 8.14T + 23T^{2} \) |
| 29 | \( 1 + 1.13T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 8.99T + 37T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + 1.77T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.32T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.862T + 89T^{2} \) |
| 97 | \( 1 + 2.95T + 97T^{2} \) |
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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
−8.196576188555642333013081976575, −7.09693591158634589835957737457, −6.53120268043903066437234111082, −5.87045995859232305746320398444, −5.12897396953256601503529642622, −4.37270744256177724989735043266, −3.58299192141551857421531208812, −3.16958807003417513503249044151, −2.04053961947627753416557312909, −1.03673219208648237607034090057,
1.03673219208648237607034090057, 2.04053961947627753416557312909, 3.16958807003417513503249044151, 3.58299192141551857421531208812, 4.37270744256177724989735043266, 5.12897396953256601503529642622, 5.87045995859232305746320398444, 6.53120268043903066437234111082, 7.09693591158634589835957737457, 8.196576188555642333013081976575