L(s) = 1 | + 5-s − 7-s + 2·11-s + 17-s + 4·19-s − 4·23-s − 4·25-s − 8·29-s − 4·31-s − 35-s − 7·37-s − 5·41-s − 43-s − 9·47-s + 49-s + 2·53-s + 2·55-s + 3·59-s − 2·61-s + 4·67-s + 12·71-s − 16·73-s − 2·77-s + 15·79-s − 3·83-s + 85-s + 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.603·11-s + 0.242·17-s + 0.917·19-s − 0.834·23-s − 4/5·25-s − 1.48·29-s − 0.718·31-s − 0.169·35-s − 1.15·37-s − 0.780·41-s − 0.152·43-s − 1.31·47-s + 1/7·49-s + 0.274·53-s + 0.269·55-s + 0.390·59-s − 0.256·61-s + 0.488·67-s + 1.42·71-s − 1.87·73-s − 0.227·77-s + 1.68·79-s − 0.329·83-s + 0.108·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.65004329832495213018292943327, −7.02149239656734196114444400280, −6.25575024261827051989994213029, −5.61510414361084546603037642384, −4.98291454268829879515680501784, −3.78536600467766840643531586830, −3.44253040392140435550826615231, −2.18628651419004802539130214164, −1.45766279791695892903734607199, 0,
1.45766279791695892903734607199, 2.18628651419004802539130214164, 3.44253040392140435550826615231, 3.78536600467766840643531586830, 4.98291454268829879515680501784, 5.61510414361084546603037642384, 6.25575024261827051989994213029, 7.02149239656734196114444400280, 7.65004329832495213018292943327