| L(s) = 1 | − 1.65·2-s + 3-s + 0.729·4-s − 1.65·6-s − 7-s + 2.09·8-s + 9-s + 11-s + 0.729·12-s − 5.75·13-s + 1.65·14-s − 4.92·16-s + 1.57·17-s − 1.65·18-s − 2.27·19-s − 21-s − 1.65·22-s − 7.38·23-s + 2.09·24-s + 9.50·26-s + 27-s − 0.729·28-s + 9.32·29-s + 5.05·31-s + 3.94·32-s + 33-s − 2.60·34-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.577·3-s + 0.364·4-s − 0.674·6-s − 0.377·7-s + 0.741·8-s + 0.333·9-s + 0.301·11-s + 0.210·12-s − 1.59·13-s + 0.441·14-s − 1.23·16-s + 0.381·17-s − 0.389·18-s − 0.520·19-s − 0.218·21-s − 0.352·22-s − 1.54·23-s + 0.428·24-s + 1.86·26-s + 0.192·27-s − 0.137·28-s + 1.73·29-s + 0.907·31-s + 0.697·32-s + 0.174·33-s − 0.446·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 + 7.38T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 8.08T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 7.48T + 97T^{2} \) |
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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
−7.86382138029420384986901050592, −7.38921022906324187376334031427, −6.61094930412702158185462736255, −5.79120339118944431277119096230, −4.51137195457408835486016266767, −4.27741338979125372142461024241, −2.87898825807306372205275408968, −2.28404660527311974361953860572, −1.16402306684476341548208949781, 0,
1.16402306684476341548208949781, 2.28404660527311974361953860572, 2.87898825807306372205275408968, 4.27741338979125372142461024241, 4.51137195457408835486016266767, 5.79120339118944431277119096230, 6.61094930412702158185462736255, 7.38921022906324187376334031427, 7.86382138029420384986901050592
