| L(s) = 1 | − 2.04e3·2-s − 2.48e5·3-s + 4.19e6·4-s + 4.88e7·5-s + 5.08e8·6-s − 5.53e9·7-s − 8.58e9·8-s − 3.26e10·9-s − 1.00e11·10-s + 1.14e12·11-s − 1.04e12·12-s + 5.00e12·13-s + 1.13e13·14-s − 1.21e13·15-s + 1.75e13·16-s + 2.66e14·17-s + 6.67e13·18-s − 8.29e14·19-s + 2.04e14·20-s + 1.37e15·21-s − 2.35e15·22-s + 4.09e15·23-s + 2.13e15·24-s + 2.38e15·25-s − 1.02e16·26-s + 3.14e16·27-s − 2.32e16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.808·3-s + 0.5·4-s + 0.447·5-s + 0.571·6-s − 1.05·7-s − 0.353·8-s − 0.346·9-s − 0.316·10-s + 1.21·11-s − 0.404·12-s + 0.774·13-s + 0.748·14-s − 0.361·15-s + 0.250·16-s + 1.88·17-s + 0.244·18-s − 1.63·19-s + 0.223·20-s + 0.855·21-s − 0.858·22-s + 0.895·23-s + 0.285·24-s + 0.200·25-s − 0.547·26-s + 1.08·27-s − 0.529·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.04e3T \) |
| 5 | \( 1 - 4.88e7T \) |
| good | 3 | \( 1 + 2.48e5T + 9.41e10T^{2} \) |
| 7 | \( 1 + 5.53e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.14e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 5.00e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.66e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 8.29e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.09e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 7.61e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 9.68e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 4.73e16T + 1.17e36T^{2} \) |
| 41 | \( 1 + 2.15e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 7.08e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.89e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 3.73e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.48e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.46e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 8.02e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 9.67e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.76e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 6.00e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.90e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 4.51e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.75e22T + 4.96e45T^{2} \) |
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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
−14.69390718322505392582060265338, −12.79049491196663800966999533364, −11.42728542323827566773725694286, −10.08346713543973632897355434531, −8.795024627342551685736053765080, −6.69201322082391387620331792881, −5.77645227090916990841145681726, −3.38027992499909036996118919657, −1.36549710130965868341011250098, 0,
1.36549710130965868341011250098, 3.38027992499909036996118919657, 5.77645227090916990841145681726, 6.69201322082391387620331792881, 8.795024627342551685736053765080, 10.08346713543973632897355434531, 11.42728542323827566773725694286, 12.79049491196663800966999533364, 14.69390718322505392582060265338
