L(s) = 1 | + 1.56·3-s + 5-s − 7-s − 0.561·9-s − 1.56·11-s + 6.68·13-s + 1.56·15-s + 7.56·17-s + 7.12·19-s − 1.56·21-s − 3.12·23-s + 25-s − 5.56·27-s + 0.438·29-s − 6.24·31-s − 2.43·33-s − 35-s − 8.24·37-s + 10.4·39-s − 1.12·41-s + 7.12·43-s − 0.561·45-s − 2.43·47-s + 49-s + 11.8·51-s − 13.1·53-s − 1.56·55-s + ⋯ |
L(s) = 1 | + 0.901·3-s + 0.447·5-s − 0.377·7-s − 0.187·9-s − 0.470·11-s + 1.85·13-s + 0.403·15-s + 1.83·17-s + 1.63·19-s − 0.340·21-s − 0.651·23-s + 0.200·25-s − 1.07·27-s + 0.0814·29-s − 1.12·31-s − 0.424·33-s − 0.169·35-s − 1.35·37-s + 1.67·39-s − 0.175·41-s + 1.08·43-s − 0.0837·45-s − 0.355·47-s + 0.142·49-s + 1.65·51-s − 1.80·53-s − 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.087324657\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087324657\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 0.684T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−10.61827957163050957265304183263, −9.738726061506570127778877462450, −9.030632724334327404026659719923, −8.146350404448586731179615719675, −7.42757827651910789407263322863, −6.00996958236054972935399296225, −5.42796671413835125869301056715, −3.59532982476847335394131149514, −3.10195565934861361687075647566, −1.48453702564647100584551910366,
1.48453702564647100584551910366, 3.10195565934861361687075647566, 3.59532982476847335394131149514, 5.42796671413835125869301056715, 6.00996958236054972935399296225, 7.42757827651910789407263322863, 8.146350404448586731179615719675, 9.030632724334327404026659719923, 9.738726061506570127778877462450, 10.61827957163050957265304183263