L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 6.74·13-s + 14-s − 15-s + 16-s − 18-s + 4.74·19-s + 20-s + 21-s + 4·22-s − 23-s + 24-s + 25-s − 6.74·26-s − 27-s − 28-s + 4.74·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.87·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 1.08·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.200·25-s − 1.32·26-s − 0.192·27-s − 0.188·28-s + 0.881·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.170466717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170466717\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 0.744T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−8.362967599396316486829863472358, −7.63434123589094092818093538131, −6.74454345873079194795529448422, −6.17685834616581263875829805446, −5.55558769398112576227433633561, −4.76968200226925766138049473349, −3.54531849207073823128693323875, −2.82252417428467463945743014812, −1.63168338707558880328145729319, −0.71065407895484290259084053375,
0.71065407895484290259084053375, 1.63168338707558880328145729319, 2.82252417428467463945743014812, 3.54531849207073823128693323875, 4.76968200226925766138049473349, 5.55558769398112576227433633561, 6.17685834616581263875829805446, 6.74454345873079194795529448422, 7.63434123589094092818093538131, 8.362967599396316486829863472358