L(s) = 1 | + 1.68·2-s − 3-s + 0.842·4-s + 3.77·5-s − 1.68·6-s − 7-s − 1.95·8-s + 9-s + 6.37·10-s − 5.93·11-s − 0.842·12-s − 0.472·13-s − 1.68·14-s − 3.77·15-s − 4.97·16-s + 6.91·17-s + 1.68·18-s − 0.471·19-s + 3.18·20-s + 21-s − 10.0·22-s + 1.57·23-s + 1.95·24-s + 9.27·25-s − 0.796·26-s − 27-s − 0.842·28-s + ⋯ |
L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.421·4-s + 1.68·5-s − 0.688·6-s − 0.377·7-s − 0.689·8-s + 0.333·9-s + 2.01·10-s − 1.79·11-s − 0.243·12-s − 0.130·13-s − 0.450·14-s − 0.975·15-s − 1.24·16-s + 1.67·17-s + 0.397·18-s − 0.108·19-s + 0.712·20-s + 0.218·21-s − 2.13·22-s + 0.329·23-s + 0.398·24-s + 1.85·25-s − 0.156·26-s − 0.192·27-s − 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.400249975\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.400249975\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 0.472T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 0.471T + 19T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + 2.77T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 + 4.80T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 0.391T + 83T^{2} \) |
| 89 | \( 1 - 2.84T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.54967063839960621028489311252, −6.87191201102512693884278757535, −5.97283945663822894488399325336, −5.58172363431548855686564062748, −5.33398414851653121050983667407, −4.58061030118477248494269111236, −3.47622927022993966989038061265, −2.74047694438372234231371941653, −2.12443962609784352131020887396, −0.76580686310005761466914101781,
0.76580686310005761466914101781, 2.12443962609784352131020887396, 2.74047694438372234231371941653, 3.47622927022993966989038061265, 4.58061030118477248494269111236, 5.33398414851653121050983667407, 5.58172363431548855686564062748, 5.97283945663822894488399325336, 6.87191201102512693884278757535, 7.54967063839960621028489311252