| L(s) = 1 | − 1.77e5·3-s − 1.74e8·5-s + 5.28e9·7-s + 3.13e10·9-s − 8.60e11·11-s − 1.20e13·13-s + 3.09e13·15-s − 1.65e14·17-s − 8.38e13·19-s − 9.36e14·21-s + 6.45e15·23-s + 1.86e16·25-s − 5.55e15·27-s + 6.43e14·29-s − 1.74e17·31-s + 1.52e17·33-s − 9.23e17·35-s − 1.06e18·37-s + 2.14e18·39-s − 4.76e18·41-s − 9.51e18·43-s − 5.48e18·45-s − 2.19e19·47-s + 5.51e17·49-s + 2.94e19·51-s + 2.92e19·53-s + 1.50e20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.60·5-s + 1.01·7-s + 0.333·9-s − 0.909·11-s − 1.87·13-s + 0.924·15-s − 1.17·17-s − 0.165·19-s − 0.583·21-s + 1.41·23-s + 1.56·25-s − 0.192·27-s + 0.00978·29-s − 1.23·31-s + 0.524·33-s − 1.61·35-s − 0.982·37-s + 1.08·39-s − 1.35·41-s − 1.56·43-s − 0.533·45-s − 1.29·47-s + 0.0201·49-s + 0.678·51-s + 0.433·53-s + 1.45·55-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
| \(L(12)\) |
\(\approx\) |
\(0.0002787830359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0002787830359\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 1.74e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 5.28e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 8.60e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.20e13T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.65e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 8.38e13T + 2.57e29T^{2} \) |
| 23 | \( 1 - 6.45e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 6.43e14T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.74e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.06e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.76e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 9.51e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.19e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 2.92e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.50e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.02e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.53e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.02e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 2.92e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.71e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 5.42e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.35e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.53e22T + 4.96e45T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.33312434886007924614738495020, −10.43610702211657812194327095332, −8.733264423267893957024090633883, −7.63991985786957450593787940299, −7.01962292939969532042045908174, −4.93541929432582584708938811994, −4.73208863587060932340650556700, −3.16865230676044069580279358435, −1.77959107699616135867409517161, −0.00726232416674873961237161583,
0.00726232416674873961237161583, 1.77959107699616135867409517161, 3.16865230676044069580279358435, 4.73208863587060932340650556700, 4.93541929432582584708938811994, 7.01962292939969532042045908174, 7.63991985786957450593787940299, 8.733264423267893957024090633883, 10.43610702211657812194327095332, 11.33312434886007924614738495020
