| L(s) = 1 | + 3.90·5-s − 36.4·7-s + 19.1·11-s + 13·13-s + 83.8·17-s − 46.8·19-s + 103.·23-s − 109.·25-s − 108.·29-s + 147.·31-s − 142.·35-s − 160.·37-s − 231.·41-s + 340.·43-s + 119.·47-s + 982.·49-s + 732.·53-s + 75.0·55-s − 229.·59-s + 108.·61-s + 50.8·65-s − 10.3·67-s − 869.·71-s − 1.09e3·73-s − 698.·77-s − 140.·79-s − 159.·83-s + ⋯ |
| L(s) = 1 | + 0.349·5-s − 1.96·7-s + 0.526·11-s + 0.277·13-s + 1.19·17-s − 0.565·19-s + 0.941·23-s − 0.877·25-s − 0.693·29-s + 0.854·31-s − 0.687·35-s − 0.710·37-s − 0.881·41-s + 1.20·43-s + 0.371·47-s + 2.86·49-s + 1.89·53-s + 0.183·55-s − 0.507·59-s + 0.228·61-s + 0.0969·65-s − 0.0189·67-s − 1.45·71-s − 1.76·73-s − 1.03·77-s − 0.199·79-s − 0.210·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 3.90T + 125T^{2} \) |
| 7 | \( 1 + 36.4T + 343T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 83.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 108.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 119.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 732.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 10.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 869.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 140.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 159.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 858.T + 9.12e5T^{2} \) |
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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
−8.784209465279355539943856661142, −7.50982501905937034483152952626, −6.82629870904303529877404574461, −6.06665053644022744229175512808, −5.54094773128734046419751771206, −4.10833508013850964561995650852, −3.40299996705130014780256046101, −2.57854078971587699878482642571, −1.17744467931919439060638322661, 0,
1.17744467931919439060638322661, 2.57854078971587699878482642571, 3.40299996705130014780256046101, 4.10833508013850964561995650852, 5.54094773128734046419751771206, 6.06665053644022744229175512808, 6.82629870904303529877404574461, 7.50982501905937034483152952626, 8.784209465279355539943856661142
