L(s) = 1 | − 2-s + 3-s + 4-s − 2.08·5-s − 6-s − 3.62·7-s − 8-s + 9-s + 2.08·10-s − 3.84·11-s + 12-s + 13-s + 3.62·14-s − 2.08·15-s + 16-s + 0.675·17-s − 18-s − 0.553·19-s − 2.08·20-s − 3.62·21-s + 3.84·22-s − 3.19·23-s − 24-s − 0.646·25-s − 26-s + 27-s − 3.62·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.933·5-s − 0.408·6-s − 1.36·7-s − 0.353·8-s + 0.333·9-s + 0.659·10-s − 1.16·11-s + 0.288·12-s + 0.277·13-s + 0.968·14-s − 0.538·15-s + 0.250·16-s + 0.163·17-s − 0.235·18-s − 0.126·19-s − 0.466·20-s − 0.790·21-s + 0.820·22-s − 0.666·23-s − 0.204·24-s − 0.129·25-s − 0.196·26-s + 0.192·27-s − 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4168509934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4168509934\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 17 | \( 1 - 0.675T + 17T^{2} \) |
| 19 | \( 1 + 0.553T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 0.912T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 + 8.18T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 3.67T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−8.009564668039008411154495756415, −7.24396969551611143770536645778, −6.72555278121970025633783421943, −5.94074090050434980571450020546, −5.04665067888658112777194072587, −3.98007841121026338010528738131, −3.34523879091290335583249565509, −2.79869175898709268660900127550, −1.77575578765183966292813506430, −0.32757127489464671459079682747,
0.32757127489464671459079682747, 1.77575578765183966292813506430, 2.79869175898709268660900127550, 3.34523879091290335583249565509, 3.98007841121026338010528738131, 5.04665067888658112777194072587, 5.94074090050434980571450020546, 6.72555278121970025633783421943, 7.24396969551611143770536645778, 8.009564668039008411154495756415