L(s) = 1 | − 2-s + 3-s + 4-s + 3.70·5-s − 6-s + 7-s − 8-s + 9-s − 3.70·10-s − 4·11-s + 12-s + 5.70·13-s − 14-s + 3.70·15-s + 16-s + 4·17-s − 18-s − 5.40·19-s + 3.70·20-s + 21-s + 4·22-s + 23-s − 24-s + 8.70·25-s − 5.70·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.17·10-s − 1.20·11-s + 0.288·12-s + 1.58·13-s − 0.267·14-s + 0.955·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 1.23·19-s + 0.827·20-s + 0.218·21-s + 0.852·22-s + 0.208·23-s − 0.204·24-s + 1.74·25-s − 1.11·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.983979540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.983979540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 29 | \( 1 - 0.298T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 + 0.298T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 1.40T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−9.953364830239368125571655074213, −9.212867373403171000311331131526, −8.405554987148443698619924414055, −7.83907588814253103422677825385, −6.53075073772413365267487489114, −5.91497445398735978294919446427, −4.94378442004194954455015025915, −3.33061698149894243859735840628, −2.26126253365041427194897828528, −1.39627103285555046757905291301,
1.39627103285555046757905291301, 2.26126253365041427194897828528, 3.33061698149894243859735840628, 4.94378442004194954455015025915, 5.91497445398735978294919446427, 6.53075073772413365267487489114, 7.83907588814253103422677825385, 8.405554987148443698619924414055, 9.212867373403171000311331131526, 9.953364830239368125571655074213