| L(s) = 1 | − 2.65e5·3-s − 1.01e8·5-s + 1.38e9·7-s − 2.38e10·9-s + 6.12e11·11-s − 5.23e12·13-s + 2.68e13·15-s − 1.41e14·17-s + 5.03e14·19-s − 3.66e14·21-s − 2.34e15·23-s − 1.66e15·25-s + 3.12e16·27-s + 4.07e15·29-s − 4.50e16·31-s − 1.62e17·33-s − 1.39e17·35-s + 5.39e17·37-s + 1.38e18·39-s − 4.92e18·41-s − 2.19e18·43-s + 2.41e18·45-s − 2.19e18·47-s − 2.54e19·49-s + 3.75e19·51-s − 2.69e18·53-s − 6.19e19·55-s + ⋯ |
| L(s) = 1 | − 0.863·3-s − 0.927·5-s + 0.264·7-s − 0.253·9-s + 0.646·11-s − 0.809·13-s + 0.801·15-s − 1.00·17-s + 0.992·19-s − 0.228·21-s − 0.512·23-s − 0.139·25-s + 1.08·27-s + 0.0620·29-s − 0.318·31-s − 0.558·33-s − 0.245·35-s + 0.498·37-s + 0.699·39-s − 1.39·41-s − 0.360·43-s + 0.235·45-s − 0.129·47-s − 0.930·49-s + 0.865·51-s − 0.0399·53-s − 0.599·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.3501317398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3501317398\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.65e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.01e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 1.38e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.12e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 5.23e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.41e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.03e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.34e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.07e15T + 4.31e33T^{2} \) |
| 31 | \( 1 + 4.50e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 5.39e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.92e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.19e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.19e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 2.69e18T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.72e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 5.47e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.95e18T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.23e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.38e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.21e22T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.07e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 8.03e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 4.40e22T + 4.96e45T^{2} \) |
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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
−11.00475958814471941979906773618, −9.661392903572709399291237238726, −8.418218316631055077621741170383, −7.35252960156552267336372536460, −6.32161221108346709560500940380, −5.13582014897626553535601206983, −4.25182338006528226224273709247, −3.02620910534835446775121186539, −1.56651796935024812935202015421, −0.25810459531142629582770567011,
0.25810459531142629582770567011, 1.56651796935024812935202015421, 3.02620910534835446775121186539, 4.25182338006528226224273709247, 5.13582014897626553535601206983, 6.32161221108346709560500940380, 7.35252960156552267336372536460, 8.418218316631055077621741170383, 9.661392903572709399291237238726, 11.00475958814471941979906773618
