| L(s) = 1 | − 3-s + 5-s + 4.73·7-s + 9-s − 4.19·11-s − 1.26·13-s − 15-s − 6.92·17-s − 19-s − 4.73·21-s + 6·23-s + 25-s − 27-s − 10.1·29-s + 2·31-s + 4.19·33-s + 4.73·35-s + 2.73·37-s + 1.26·39-s + 11.6·41-s + 4.73·43-s + 45-s + 10·47-s + 15.3·49-s + 6.92·51-s − 4.19·55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.78·7-s + 0.333·9-s − 1.26·11-s − 0.351·13-s − 0.258·15-s − 1.68·17-s − 0.229·19-s − 1.03·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s − 1.89·29-s + 0.359·31-s + 0.730·33-s + 0.799·35-s + 0.449·37-s + 0.203·39-s + 1.82·41-s + 0.721·43-s + 0.149·45-s + 1.45·47-s + 2.19·49-s + 0.970·51-s − 0.565·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840002862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840002862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−8.328761330012303401628125071910, −7.45746371869388320442434644017, −7.10261496941497293184373385082, −5.87384596442732967011460762441, −5.38297948864944214045115863347, −4.70702086601384250192774259482, −4.14630576128860509960993824945, −2.49300375073771484602837124552, −2.05706948783891339377207549974, −0.77798017491722243677583252580,
0.77798017491722243677583252580, 2.05706948783891339377207549974, 2.49300375073771484602837124552, 4.14630576128860509960993824945, 4.70702086601384250192774259482, 5.38297948864944214045115863347, 5.87384596442732967011460762441, 7.10261496941497293184373385082, 7.45746371869388320442434644017, 8.328761330012303401628125071910
