| L(s) = 1 | + 3·3-s + 3.46·5-s − 24.2·7-s + 9·9-s + 48·11-s + 41.5·13-s + 10.3·15-s + 54·17-s − 4·19-s − 72.7·21-s − 173.·23-s − 113·25-s + 27·27-s + 162.·29-s + 58.8·31-s + 144·33-s − 84·35-s − 325.·37-s + 124.·39-s + 294·41-s + 188·43-s + 31.1·45-s + 505.·47-s + 245·49-s + 162·51-s + 744.·53-s + 166.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.309·5-s − 1.30·7-s + 0.333·9-s + 1.31·11-s + 0.886·13-s + 0.178·15-s + 0.770·17-s − 0.0482·19-s − 0.755·21-s − 1.57·23-s − 0.904·25-s + 0.192·27-s + 1.04·29-s + 0.341·31-s + 0.759·33-s − 0.405·35-s − 1.44·37-s + 0.512·39-s + 1.11·41-s + 0.666·43-s + 0.103·45-s + 1.56·47-s + 0.714·49-s + 0.444·51-s + 1.93·53-s + 0.407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.618928687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.618928687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 - 3.46T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 - 48T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188T + 7.95e4T^{2} \) |
| 47 | \( 1 - 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 252T + 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 - 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−9.864213414231707513735628758823, −9.127554318090184601681127317662, −8.402222838822267796650617855968, −7.27034818232677890748115266246, −6.34838347596536759967615958940, −5.79600529257246557483843267720, −4.03561332930643849414782087242, −3.55236082540374648393595097664, −2.26592253153613633575974834015, −0.916964568446941825513274142864,
0.916964568446941825513274142864, 2.26592253153613633575974834015, 3.55236082540374648393595097664, 4.03561332930643849414782087242, 5.79600529257246557483843267720, 6.34838347596536759967615958940, 7.27034818232677890748115266246, 8.402222838822267796650617855968, 9.127554318090184601681127317662, 9.864213414231707513735628758823
