| L(s) = 1 | + 1.83·2-s + 1.36·4-s − 7-s − 1.16·8-s − 2.19·11-s + 5.03·13-s − 1.83·14-s − 4.86·16-s + 3.23·17-s − 6.43·19-s − 4.03·22-s − 0.364·23-s + 9.23·26-s − 1.36·28-s + 0.364·29-s − 8.70·31-s − 6.59·32-s + 5.92·34-s + 2.76·37-s − 11.7·38-s − 7.32·41-s + 9.19·43-s − 3.00·44-s − 0.668·46-s − 10.7·47-s + 49-s + 6.86·52-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.682·4-s − 0.377·7-s − 0.412·8-s − 0.662·11-s + 1.39·13-s − 0.490·14-s − 1.21·16-s + 0.783·17-s − 1.47·19-s − 0.859·22-s − 0.0759·23-s + 1.81·26-s − 0.257·28-s + 0.0676·29-s − 1.56·31-s − 1.16·32-s + 1.01·34-s + 0.454·37-s − 1.91·38-s − 1.14·41-s + 1.40·43-s − 0.452·44-s − 0.0985·46-s − 1.56·47-s + 0.142·49-s + 0.952·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 5.03T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 + 0.364T + 23T^{2} \) |
| 29 | \( 1 - 0.364T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 9.19T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 0.740T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 + 7.99T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.87981219100028227172204537533, −6.94299011154861351159404329458, −6.13408964004908634789885468252, −5.79931276103549145285478015829, −4.93557470429255057909127258877, −4.12352165809774644756744980007, −3.50277931449493717046875164951, −2.78408521181444822526835171788, −1.66379201495422729046026648013, 0,
1.66379201495422729046026648013, 2.78408521181444822526835171788, 3.50277931449493717046875164951, 4.12352165809774644756744980007, 4.93557470429255057909127258877, 5.79931276103549145285478015829, 6.13408964004908634789885468252, 6.94299011154861351159404329458, 7.87981219100028227172204537533
