L(s) = 1 | + 2.09·2-s − 0.510·3-s + 2.37·4-s + 4.43·5-s − 1.06·6-s − 2.30·7-s + 0.777·8-s − 2.73·9-s + 9.28·10-s + 5.75·11-s − 1.21·12-s − 13-s − 4.81·14-s − 2.26·15-s − 3.11·16-s + 0.581·17-s − 5.72·18-s − 3.67·19-s + 10.5·20-s + 1.17·21-s + 12.0·22-s − 7.44·23-s − 0.396·24-s + 14.7·25-s − 2.09·26-s + 2.92·27-s − 5.45·28-s + ⋯ |
L(s) = 1 | + 1.47·2-s − 0.294·3-s + 1.18·4-s + 1.98·5-s − 0.435·6-s − 0.869·7-s + 0.274·8-s − 0.913·9-s + 2.93·10-s + 1.73·11-s − 0.349·12-s − 0.277·13-s − 1.28·14-s − 0.585·15-s − 0.779·16-s + 0.140·17-s − 1.35·18-s − 0.842·19-s + 2.35·20-s + 0.256·21-s + 2.56·22-s − 1.55·23-s − 0.0810·24-s + 2.94·25-s − 0.410·26-s + 0.563·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.103067608\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.103067608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 3 | \( 1 + 0.510T + 3T^{2} \) |
| 5 | \( 1 - 4.43T + 5T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 - 5.75T + 11T^{2} \) |
| 17 | \( 1 - 0.581T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 - 0.352T + 43T^{2} \) |
| 47 | \( 1 - 0.562T + 47T^{2} \) |
| 53 | \( 1 - 7.57T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 1.24T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 4.04T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 0.601T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−11.63447411064869357848869510795, −10.40910668550558558986562612305, −9.509369841448649064068575463423, −8.850679235539087753465053195725, −6.66354744683991344018259535940, −6.16561737271235110895209982530, −5.73323691370924331394258825212, −4.47234348554627009813807443360, −3.17587029915480142777935114071, −2.03133014690974556374267189730,
2.03133014690974556374267189730, 3.17587029915480142777935114071, 4.47234348554627009813807443360, 5.73323691370924331394258825212, 6.16561737271235110895209982530, 6.66354744683991344018259535940, 8.850679235539087753465053195725, 9.509369841448649064068575463423, 10.40910668550558558986562612305, 11.63447411064869357848869510795