| L(s) = 1 | − 2.12·2-s − 3-s + 2.52·4-s + 5-s + 2.12·6-s − 2.46·7-s − 1.11·8-s + 9-s − 2.12·10-s − 2.49·11-s − 2.52·12-s + 0.527·13-s + 5.24·14-s − 15-s − 2.67·16-s − 4.57·17-s − 2.12·18-s − 5.03·19-s + 2.52·20-s + 2.46·21-s + 5.31·22-s + 1.11·24-s + 25-s − 1.12·26-s − 27-s − 6.22·28-s + 1.30·29-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.577·3-s + 1.26·4-s + 0.447·5-s + 0.868·6-s − 0.931·7-s − 0.395·8-s + 0.333·9-s − 0.672·10-s − 0.753·11-s − 0.729·12-s + 0.146·13-s + 1.40·14-s − 0.258·15-s − 0.668·16-s − 1.10·17-s − 0.501·18-s − 1.15·19-s + 0.564·20-s + 0.537·21-s + 1.13·22-s + 0.228·24-s + 0.200·25-s − 0.219·26-s − 0.192·27-s − 1.17·28-s + 0.242·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 - 0.527T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + 1.05T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 - 6.45T + 61T^{2} \) |
| 67 | \( 1 - 8.91T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 0.703T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−7.57222094718469030622590033667, −6.81592878932667913804725211245, −6.39218225542060888898204065165, −5.69922199754320367378340077245, −4.71766067764799335800846798574, −3.94040136638440342686759001218, −2.60449294201606310342332792384, −2.11625763475859729002353720165, −0.861237398389777400742820790250, 0,
0.861237398389777400742820790250, 2.11625763475859729002353720165, 2.60449294201606310342332792384, 3.94040136638440342686759001218, 4.71766067764799335800846798574, 5.69922199754320367378340077245, 6.39218225542060888898204065165, 6.81592878932667913804725211245, 7.57222094718469030622590033667
