| L(s) = 1 | − 2.61·3-s − 1.23·5-s − 3·7-s + 3.85·9-s − 0.618·11-s − 13-s + 3.23·15-s + 5.23·17-s + 7.85·21-s − 7.61·23-s − 3.47·25-s − 2.23·27-s − 1.38·29-s + 2.14·31-s + 1.61·33-s + 3.70·35-s + 2.14·37-s + 2.61·39-s − 3·41-s + 6.85·43-s − 4.76·45-s − 3·47-s + 2·49-s − 13.7·51-s + 9.32·53-s + 0.763·55-s + 15.3·59-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 0.552·5-s − 1.13·7-s + 1.28·9-s − 0.186·11-s − 0.277·13-s + 0.835·15-s + 1.26·17-s + 1.71·21-s − 1.58·23-s − 0.694·25-s − 0.430·27-s − 0.256·29-s + 0.385·31-s + 0.281·33-s + 0.626·35-s + 0.352·37-s + 0.419·39-s − 0.468·41-s + 1.04·43-s − 0.710·45-s − 0.437·47-s + 0.285·49-s − 1.91·51-s + 1.28·53-s + 0.103·55-s + 1.99·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 0.618T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 - 15.3T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.472T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−7.63751817737909277209078283548, −6.90630907662818607417647333284, −6.19132751398449113348874735912, −5.73082788026410644270678688340, −5.04929037281067584551559608483, −4.05086223462941059673015450705, −3.50193619472823696350193212217, −2.30626639995931861879522495571, −0.863803242471009052766916878900, 0,
0.863803242471009052766916878900, 2.30626639995931861879522495571, 3.50193619472823696350193212217, 4.05086223462941059673015450705, 5.04929037281067584551559608483, 5.73082788026410644270678688340, 6.19132751398449113348874735912, 6.90630907662818607417647333284, 7.63751817737909277209078283548
