L(s) = 1 | + 2-s − 2.45·3-s + 4-s − 5-s − 2.45·6-s + 3.60·7-s + 8-s + 3.03·9-s − 10-s + 3.51·11-s − 2.45·12-s − 0.574·13-s + 3.60·14-s + 2.45·15-s + 16-s + 0.123·17-s + 3.03·18-s + 5.64·19-s − 20-s − 8.85·21-s + 3.51·22-s + 7.23·23-s − 2.45·24-s + 25-s − 0.574·26-s − 0.0861·27-s + 3.60·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.447·5-s − 1.00·6-s + 1.36·7-s + 0.353·8-s + 1.01·9-s − 0.316·10-s + 1.05·11-s − 0.709·12-s − 0.159·13-s + 0.962·14-s + 0.634·15-s + 0.250·16-s + 0.0300·17-s + 0.715·18-s + 1.29·19-s − 0.223·20-s − 1.93·21-s + 0.749·22-s + 1.50·23-s − 0.501·24-s + 0.200·25-s − 0.112·26-s − 0.0165·27-s + 0.680·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.478987876\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.478987876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 + 0.574T + 13T^{2} \) |
| 17 | \( 1 - 0.123T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.558T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 3.93T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 + 1.35T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−7.71049435561521008043477155321, −7.24847535339990494407360596641, −6.58025850916098609525377710717, −5.66988056750476943914477054428, −5.25370907209886734651130841698, −4.60288098988664106960379363247, −3.98432549228827802567983364267, −2.94537608652020617517371404438, −1.57816258426097262844861789382, −0.888535615890531926373412378792,
0.888535615890531926373412378792, 1.57816258426097262844861789382, 2.94537608652020617517371404438, 3.98432549228827802567983364267, 4.60288098988664106960379363247, 5.25370907209886734651130841698, 5.66988056750476943914477054428, 6.58025850916098609525377710717, 7.24847535339990494407360596641, 7.71049435561521008043477155321