| L(s) = 1 | − 3.43·5-s − 7-s − 2.61·11-s + 5.43·13-s + 0.611·17-s − 19-s + 1.43·23-s + 6.79·25-s − 1.74·29-s + 0.255·31-s + 3.43·35-s + 5.79·37-s − 8.96·41-s + 7.58·43-s + 10.6·47-s + 49-s − 5.53·53-s + 8.96·55-s − 4.30·59-s − 1.27·61-s − 18.6·65-s + 0.611·67-s + 1.07·71-s − 11.6·73-s + 2.61·77-s − 12.4·79-s + 10.4·83-s + ⋯ |
| L(s) = 1 | − 1.53·5-s − 0.377·7-s − 0.787·11-s + 1.50·13-s + 0.148·17-s − 0.229·19-s + 0.298·23-s + 1.35·25-s − 0.323·29-s + 0.0458·31-s + 0.580·35-s + 0.951·37-s − 1.40·41-s + 1.15·43-s + 1.55·47-s + 0.142·49-s − 0.760·53-s + 1.20·55-s − 0.559·59-s − 0.163·61-s − 2.31·65-s + 0.0747·67-s + 0.127·71-s − 1.36·73-s + 0.297·77-s − 1.40·79-s + 1.14·83-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 | \( 1 + 3.43T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 - 0.611T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 0.255T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 1.27T + 61T^{2} \) |
| 67 | \( 1 - 0.611T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−7.80233357009246873797216254831, −7.45709637976400741588544457565, −6.48949630681350100625244847123, −5.81180814346290804813736336848, −4.83133918485081641807896272980, −4.01055706408249825884400235926, −3.48509581821501164855341846140, −2.63091160861477335470633531621, −1.14207886106608566229236838372, 0,
1.14207886106608566229236838372, 2.63091160861477335470633531621, 3.48509581821501164855341846140, 4.01055706408249825884400235926, 4.83133918485081641807896272980, 5.81180814346290804813736336848, 6.48949630681350100625244847123, 7.45709637976400741588544457565, 7.80233357009246873797216254831
