L(s) = 1 | − 2-s + 4-s − 2.26·5-s − 0.397·7-s − 8-s + 2.26·10-s + 3.96·11-s + 0.823·13-s + 0.397·14-s + 16-s − 6.85·17-s − 8.38·19-s − 2.26·20-s − 3.96·22-s + 8.28·23-s + 0.142·25-s − 0.823·26-s − 0.397·28-s + 8.78·29-s + 4.00·31-s − 32-s + 6.85·34-s + 0.902·35-s + 0.694·37-s + 8.38·38-s + 2.26·40-s + 9.11·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.01·5-s − 0.150·7-s − 0.353·8-s + 0.717·10-s + 1.19·11-s + 0.228·13-s + 0.106·14-s + 0.250·16-s − 1.66·17-s − 1.92·19-s − 0.507·20-s − 0.846·22-s + 1.72·23-s + 0.0284·25-s − 0.161·26-s − 0.0751·28-s + 1.63·29-s + 0.719·31-s − 0.176·32-s + 1.17·34-s + 0.152·35-s + 0.114·37-s + 1.36·38-s + 0.358·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8620038322\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8620038322\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 + 0.397T + 7T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 - 0.823T + 13T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 + 8.38T + 19T^{2} \) |
| 23 | \( 1 - 8.28T + 23T^{2} \) |
| 29 | \( 1 - 8.78T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 0.694T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−8.096678252610705263237505235399, −7.05656403793048832062720605146, −6.50679236300598583076816561127, −6.27386213145412727571797457639, −4.62171252912898233233221690437, −4.45234736474307324411243657620, −3.45667233954838104650042714194, −2.62781327873148395839399223149, −1.60087766238926537706458266803, −0.51864563540575480112053900073,
0.51864563540575480112053900073, 1.60087766238926537706458266803, 2.62781327873148395839399223149, 3.45667233954838104650042714194, 4.45234736474307324411243657620, 4.62171252912898233233221690437, 6.27386213145412727571797457639, 6.50679236300598583076816561127, 7.05656403793048832062720605146, 8.096678252610705263237505235399