| L(s) = 1 | + 1.77e5·3-s + 1.13e8·5-s − 7.85e9·7-s + 3.13e10·9-s − 1.03e12·11-s + 8.08e12·13-s + 2.01e13·15-s − 1.38e14·17-s + 1.41e14·19-s − 1.39e15·21-s − 4.80e15·23-s + 9.90e14·25-s + 5.55e15·27-s − 1.45e16·29-s + 8.10e16·31-s − 1.82e17·33-s − 8.92e17·35-s + 1.19e18·37-s + 1.43e18·39-s + 6.63e18·41-s + 1.06e19·43-s + 3.56e18·45-s − 1.16e19·47-s + 3.42e19·49-s − 2.45e19·51-s − 7.56e19·53-s − 1.17e20·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.04·5-s − 1.50·7-s + 0.333·9-s − 1.08·11-s + 1.25·13-s + 0.600·15-s − 0.981·17-s + 0.279·19-s − 0.866·21-s − 1.05·23-s + 0.0831·25-s + 0.192·27-s − 0.221·29-s + 0.573·31-s − 0.629·33-s − 1.56·35-s + 1.10·37-s + 0.722·39-s + 1.88·41-s + 1.74·43-s + 0.346·45-s − 0.690·47-s + 1.25·49-s − 0.566·51-s − 1.12·53-s − 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(2.573024940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573024940\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.77e5T \) |
| good | 5 | \( 1 - 1.13e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 7.85e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.03e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 8.08e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.38e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 1.41e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 4.80e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.45e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 8.10e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.19e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 6.63e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.06e19T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.16e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 7.56e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 4.19e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 7.45e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.29e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.37e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.03e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.75e20T + 4.42e43T^{2} \) |
| 83 | \( 1 - 9.99e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 5.36e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.11e22T + 4.96e45T^{2} \) |
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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
−10.92160850552694720830548800209, −9.857591934590609998847470903188, −9.200418774869806978166330099075, −7.907374377206360660608684178491, −6.43862408548081759835405084129, −5.79677441338877949979480284066, −4.09329801952537671502642941657, −2.90372391454675358721395006982, −2.12201807870852333761733988358, −0.65931674201393174417122892785,
0.65931674201393174417122892785, 2.12201807870852333761733988358, 2.90372391454675358721395006982, 4.09329801952537671502642941657, 5.79677441338877949979480284066, 6.43862408548081759835405084129, 7.907374377206360660608684178491, 9.200418774869806978166330099075, 9.857591934590609998847470903188, 10.92160850552694720830548800209
