| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 3·11-s + 3·13-s + 15-s + 17-s + 6·19-s − 21-s + 2·23-s − 4·25-s − 27-s − 6·29-s − 4·31-s − 3·33-s − 35-s + 11·37-s − 3·39-s + 12·41-s + 3·43-s − 45-s − 12·47-s + 49-s − 51-s − 5·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.832·13-s + 0.258·15-s + 0.242·17-s + 1.37·19-s − 0.218·21-s + 0.417·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.169·35-s + 1.80·37-s − 0.480·39-s + 1.87·41-s + 0.457·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.140·51-s − 0.686·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.201682089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.201682089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.72467999913062, −14.82377107058196, −14.50308671143560, −13.96953957789368, −13.08227295249546, −12.93631993376034, −12.01749897362750, −11.62638215770649, −11.06847882426592, −10.98919140870228, −9.785352327008006, −9.511090057037351, −8.955944062877241, −7.969381863067396, −7.747285524150375, −7.050338451392717, −6.315661174022477, −5.802201071780280, −5.220403394712505, −4.434883438003452, −3.808232371442687, −3.344886740077567, −2.209432816893034, −1.305677045186336, −0.7037223300463442,
0.7037223300463442, 1.305677045186336, 2.209432816893034, 3.344886740077567, 3.808232371442687, 4.434883438003452, 5.220403394712505, 5.802201071780280, 6.315661174022477, 7.050338451392717, 7.747285524150375, 7.969381863067396, 8.955944062877241, 9.511090057037351, 9.785352327008006, 10.98919140870228, 11.06847882426592, 11.62638215770649, 12.01749897362750, 12.93631993376034, 13.08227295249546, 13.96953957789368, 14.50308671143560, 14.82377107058196, 15.72467999913062
