| L(s) = 1 | − 2.39·2-s − 3-s + 3.73·4-s + 1.50·5-s + 2.39·6-s − 1.58·7-s − 4.15·8-s + 9-s − 3.60·10-s + 2.71·11-s − 3.73·12-s + 5.17·13-s + 3.78·14-s − 1.50·15-s + 2.47·16-s + 1.40·17-s − 2.39·18-s + 1.24·19-s + 5.62·20-s + 1.58·21-s − 6.50·22-s + 7.32·23-s + 4.15·24-s − 2.73·25-s − 12.3·26-s − 27-s − 5.90·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.86·4-s + 0.673·5-s + 0.977·6-s − 0.597·7-s − 1.46·8-s + 0.333·9-s − 1.14·10-s + 0.819·11-s − 1.07·12-s + 1.43·13-s + 1.01·14-s − 0.388·15-s + 0.618·16-s + 0.340·17-s − 0.564·18-s + 0.285·19-s + 1.25·20-s + 0.344·21-s − 1.38·22-s + 1.52·23-s + 0.847·24-s − 0.546·25-s − 2.42·26-s − 0.192·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2253 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8153867823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8153867823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 - 0.566T + 29T^{2} \) |
| 31 | \( 1 - 0.482T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 5.76T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 + 0.0220T + 61T^{2} \) |
| 67 | \( 1 + 5.11T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.179415703337273454267390470202, −8.511078829335422266166519414171, −7.55045884358530939139500308484, −6.73048944445197083947518377013, −6.25266655517754619717805735361, −5.46867909071357780669921826876, −4.03629414066546019639354147928, −2.88742297595206134271854828274, −1.57487482721815599245164446123, −0.852774142856779805442300333030,
0.852774142856779805442300333030, 1.57487482721815599245164446123, 2.88742297595206134271854828274, 4.03629414066546019639354147928, 5.46867909071357780669921826876, 6.25266655517754619717805735361, 6.73048944445197083947518377013, 7.55045884358530939139500308484, 8.511078829335422266166519414171, 9.179415703337273454267390470202
