| L(s) = 1 | + 2.37·5-s − 7-s − 1.37·13-s − 17-s − 6.74·19-s − 5.37·23-s + 0.627·25-s − 0.627·29-s − 2.62·31-s − 2.37·35-s + 4.37·37-s − 5.11·41-s − 7.74·43-s + 0.372·47-s + 49-s − 12.1·53-s − 0.255·59-s + 1.25·61-s − 3.25·65-s − 4.62·67-s − 7.37·71-s + 10·73-s + 9.11·79-s + 15.8·83-s − 2.37·85-s − 2.62·89-s + 1.37·91-s + ⋯ |
| L(s) = 1 | + 1.06·5-s − 0.377·7-s − 0.380·13-s − 0.242·17-s − 1.54·19-s − 1.12·23-s + 0.125·25-s − 0.116·29-s − 0.471·31-s − 0.400·35-s + 0.718·37-s − 0.799·41-s − 1.18·43-s + 0.0543·47-s + 0.142·49-s − 1.66·53-s − 0.0332·59-s + 0.160·61-s − 0.403·65-s − 0.565·67-s − 0.874·71-s + 1.17·73-s + 1.02·79-s + 1.74·83-s − 0.257·85-s − 0.278·89-s + 0.143·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2.37T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 0.627T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 - 0.372T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.255T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 + 4.62T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−8.398768393068812156685188167041, −7.64825102058486466637990604626, −6.50031572107812911939072550215, −6.28006438265669109554190150660, −5.33791860142897426490875780476, −4.48884480564001278969219323941, −3.53118447055816995864215093944, −2.37907167976985434292896947456, −1.75018176920139752844312177326, 0,
1.75018176920139752844312177326, 2.37907167976985434292896947456, 3.53118447055816995864215093944, 4.48884480564001278969219323941, 5.33791860142897426490875780476, 6.28006438265669109554190150660, 6.50031572107812911939072550215, 7.64825102058486466637990604626, 8.398768393068812156685188167041
