L(s) = 1 | + 1.41i·5-s + i·7-s + 1.41·11-s + 1.41i·17-s − 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s + 1.41i·41-s − 49-s + 2.00i·55-s + 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + i·7-s + 1.41·11-s + 1.41i·17-s − 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s + 1.41i·41-s − 49-s + 2.00i·55-s + 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.183122469\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183122469\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.525831406850213719209529330084, −8.796876090294785283832789724259, −8.024783850143527977247641519413, −6.95067350979800952610770344633, −6.41620939846779624137200615489, −5.87805167280411652657719349527, −4.57235103219738370142888263103, −3.62851429915875349752060224539, −2.75275375171096306086207964270, −1.83053504815339100853283845117,
0.911336349285537840221428926381, 1.85049143501637111046529227519, 3.69769869507050417664612806438, 4.10731287785860528203320251107, 5.04820317817149915537790032745, 5.91417226871951285500437089425, 6.81602805952662671527989733065, 7.72795850514979939633403928328, 8.338299410909366001810149279223, 9.243428147065903774952406887985