| L(s) = 1 | − 2.73·3-s + 5-s − 3.46·7-s + 4.46·9-s − 4.73·11-s − 13-s − 2.73·15-s + 3.46·17-s − 3.26·19-s + 9.46·21-s + 8.19·23-s + 25-s − 3.99·27-s + 5.46·29-s + 4.73·31-s + 12.9·33-s − 3.46·35-s + 2.92·37-s + 2.73·39-s − 11.4·41-s − 2.73·43-s + 4.46·45-s + 11.4·47-s + 4.99·49-s − 9.46·51-s − 11.4·53-s − 4.73·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s − 1.30·7-s + 1.48·9-s − 1.42·11-s − 0.277·13-s − 0.705·15-s + 0.840·17-s − 0.749·19-s + 2.06·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s + 1.01·29-s + 0.849·31-s + 2.25·33-s − 0.585·35-s + 0.481·37-s + 0.437·39-s − 1.79·41-s − 0.416·43-s + 0.665·45-s + 1.67·47-s + 0.714·49-s − 1.32·51-s − 1.57·53-s − 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 1.80T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.924556127440171673570594437290, −6.92297408956779928419352381173, −6.56278735134248137142832891559, −5.80890462462597810055224435736, −5.18324571691467376107017036004, −4.64062699937574084895990812403, −3.28537677782255646499112458168, −2.55562416575041674176406491642, −1.00258231606128310147078879795, 0,
1.00258231606128310147078879795, 2.55562416575041674176406491642, 3.28537677782255646499112458168, 4.64062699937574084895990812403, 5.18324571691467376107017036004, 5.80890462462597810055224435736, 6.56278735134248137142832891559, 6.92297408956779928419352381173, 7.924556127440171673570594437290
