| L(s) = 1 | − 1.82·3-s − 4.05·5-s + 0.493·7-s + 0.313·9-s − 0.908·11-s + 5.79·13-s + 7.38·15-s − 2·17-s − 5.01·19-s − 0.897·21-s + 8.81·23-s + 11.4·25-s + 4.89·27-s − 2.70·29-s + 3.58·31-s + 1.65·33-s − 2·35-s − 37-s − 10.5·39-s − 7.23·41-s + 11.3·43-s − 1.27·45-s − 0.965·47-s − 6.75·49-s + 3.64·51-s − 3.72·53-s + 3.68·55-s + ⋯ |
| L(s) = 1 | − 1.05·3-s − 1.81·5-s + 0.186·7-s + 0.104·9-s − 0.274·11-s + 1.60·13-s + 1.90·15-s − 0.485·17-s − 1.15·19-s − 0.195·21-s + 1.83·23-s + 2.29·25-s + 0.941·27-s − 0.501·29-s + 0.643·31-s + 0.287·33-s − 0.338·35-s − 0.164·37-s − 1.69·39-s − 1.13·41-s + 1.73·43-s − 0.189·45-s − 0.140·47-s − 0.965·49-s + 0.509·51-s − 0.512·53-s + 0.497·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 + 4.05T + 5T^{2} \) |
| 7 | \( 1 - 0.493T + 7T^{2} \) |
| 11 | \( 1 + 0.908T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.965T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.96T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 + 0.444T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−8.469480331801969638369370643396, −7.928615392107728752615898700183, −6.87381008658901269801176765718, −6.42397260714471850444694819540, −5.34066854808587292445393622456, −4.57280645085264716298284978944, −3.85686218713175372285096889502, −2.93802876747073756415916636238, −1.09762550958430983904009735136, 0,
1.09762550958430983904009735136, 2.93802876747073756415916636238, 3.85686218713175372285096889502, 4.57280645085264716298284978944, 5.34066854808587292445393622456, 6.42397260714471850444694819540, 6.87381008658901269801176765718, 7.928615392107728752615898700183, 8.469480331801969638369370643396
