| L(s) = 1 | − 0.566·2-s − 3.10·3-s − 1.67·4-s + 1.75·6-s + 3.12·7-s + 2.08·8-s + 6.63·9-s − 1.90·11-s + 5.21·12-s − 5.79·13-s − 1.76·14-s + 2.18·16-s + 7.75·17-s − 3.75·18-s − 9.68·21-s + 1.07·22-s − 1.43·23-s − 6.46·24-s + 3.28·26-s − 11.2·27-s − 5.24·28-s − 1.18·29-s + 0.520·31-s − 5.39·32-s + 5.91·33-s − 4.39·34-s − 11.1·36-s + ⋯ |
| L(s) = 1 | − 0.400·2-s − 1.79·3-s − 0.839·4-s + 0.717·6-s + 1.17·7-s + 0.736·8-s + 2.21·9-s − 0.575·11-s + 1.50·12-s − 1.60·13-s − 0.472·14-s + 0.545·16-s + 1.88·17-s − 0.884·18-s − 2.11·21-s + 0.230·22-s − 0.298·23-s − 1.31·24-s + 0.643·26-s − 2.16·27-s − 0.990·28-s − 0.219·29-s + 0.0934·31-s − 0.954·32-s + 1.03·33-s − 0.752·34-s − 1.85·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7060678438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7060678438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.566T + 2T^{2} \) |
| 3 | \( 1 + 3.10T + 3T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 0.520T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 6.25T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 0.458T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 + 1.87T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 6.31T + 71T^{2} \) |
| 73 | \( 1 - 5.77T + 73T^{2} \) |
| 79 | \( 1 + 6.02T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 5.42T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−7.75605860136811350516753581121, −7.27812993777031123427810473494, −6.13715449834921812022058020297, −5.49119268424918247935656515508, −5.00460710917748781851682514908, −4.65880039168084256182911447776, −3.80912706120628350149036214818, −2.35411158721323157922186724485, −1.21855642666742108375815001814, −0.57059123299964329811323315005,
0.57059123299964329811323315005, 1.21855642666742108375815001814, 2.35411158721323157922186724485, 3.80912706120628350149036214818, 4.65880039168084256182911447776, 5.00460710917748781851682514908, 5.49119268424918247935656515508, 6.13715449834921812022058020297, 7.27812993777031123427810473494, 7.75605860136811350516753581121
