L(s) = 1 | + 0.786·3-s + 3.29·5-s − 2.08·7-s − 2.38·9-s + 1.29·11-s + 1.21·13-s + 2.59·15-s + 4.08·17-s − 19-s − 1.63·21-s − 8.95·23-s + 5.87·25-s − 4.23·27-s − 9.38·29-s + 1.02·33-s − 6.87·35-s − 2·37-s + 0.954·39-s + 3.57·41-s + 7.72·43-s − 7.85·45-s + 9.46·47-s − 2.65·49-s + 3.21·51-s − 11.9·53-s + 4.27·55-s − 0.786·57-s + ⋯ |
L(s) = 1 | + 0.454·3-s + 1.47·5-s − 0.787·7-s − 0.793·9-s + 0.391·11-s + 0.336·13-s + 0.669·15-s + 0.990·17-s − 0.229·19-s − 0.357·21-s − 1.86·23-s + 1.17·25-s − 0.814·27-s − 1.74·29-s + 0.177·33-s − 1.16·35-s − 0.328·37-s + 0.152·39-s + 0.558·41-s + 1.17·43-s − 1.17·45-s + 1.38·47-s − 0.379·49-s + 0.449·51-s − 1.64·53-s + 0.576·55-s − 0.104·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.397452702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397452702\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 0.786T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 23 | \( 1 + 8.95T + 23T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−13.17205987899768735197348638177, −12.17906874774216255739435388579, −10.79193256518609935022383920031, −9.672178196414392441713794897726, −9.194008605741772733426578296264, −7.84279750356026445851274307836, −6.22606288960254609586666388101, −5.67343847485789525497648208583, −3.58345035991103604517941245440, −2.13577551801823398844899914823,
2.13577551801823398844899914823, 3.58345035991103604517941245440, 5.67343847485789525497648208583, 6.22606288960254609586666388101, 7.84279750356026445851274307836, 9.194008605741772733426578296264, 9.672178196414392441713794897726, 10.79193256518609935022383920031, 12.17906874774216255739435388579, 13.17205987899768735197348638177