| L(s) = 1 | − 3·5-s + 7-s − 0.791·11-s − 13-s + 0.791·17-s + 19-s + 4.58·23-s + 4·25-s + 0.791·29-s − 6.37·31-s − 3·35-s + 5·37-s − 0.791·41-s + 2·43-s − 1.41·47-s + 49-s − 5.37·53-s + 2.37·55-s + 6.16·59-s − 61-s + 3·65-s + 4.37·67-s − 6.16·71-s + 2.62·73-s − 0.791·77-s − 10·79-s − 0.626·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.238·11-s − 0.277·13-s + 0.191·17-s + 0.229·19-s + 0.955·23-s + 0.800·25-s + 0.146·29-s − 1.14·31-s − 0.507·35-s + 0.821·37-s − 0.123·41-s + 0.304·43-s − 0.206·47-s + 0.142·49-s − 0.738·53-s + 0.320·55-s + 0.802·59-s − 0.128·61-s + 0.372·65-s + 0.534·67-s − 0.731·71-s + 0.307·73-s − 0.0901·77-s − 1.12·79-s − 0.0687·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.791T + 17T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 - 0.791T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 0.791T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 4.37T + 67T^{2} \) |
| 71 | \( 1 + 6.16T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 0.626T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.80283762498430611359299703056, −7.40759584338356605845797064655, −6.66512922165972716841209446383, −5.59300398234536532776870612585, −4.89136019623863065399816131807, −4.14791713355486299485805038780, −3.41875162218618716461444116800, −2.54451736757246010219507418840, −1.22594558923348279748418013920, 0,
1.22594558923348279748418013920, 2.54451736757246010219507418840, 3.41875162218618716461444116800, 4.14791713355486299485805038780, 4.89136019623863065399816131807, 5.59300398234536532776870612585, 6.66512922165972716841209446383, 7.40759584338356605845797064655, 7.80283762498430611359299703056
