L(s) = 1 | − 2.60·2-s + 2.93·3-s + 4.80·4-s − 7.64·6-s + 7-s − 7.30·8-s + 5.58·9-s + 0.264·11-s + 14.0·12-s + 1.35·13-s − 2.60·14-s + 9.45·16-s + 5.30·17-s − 14.5·18-s − 4.34·19-s + 2.93·21-s − 0.690·22-s + 23-s − 21.4·24-s − 3.53·26-s + 7.57·27-s + 4.80·28-s − 5.48·29-s + 8.13·31-s − 10.0·32-s + 0.776·33-s − 13.8·34-s + ⋯ |
L(s) = 1 | − 1.84·2-s + 1.69·3-s + 2.40·4-s − 3.11·6-s + 0.377·7-s − 2.58·8-s + 1.86·9-s + 0.0798·11-s + 4.06·12-s + 0.376·13-s − 0.697·14-s + 2.36·16-s + 1.28·17-s − 3.43·18-s − 0.995·19-s + 0.639·21-s − 0.147·22-s + 0.208·23-s − 4.37·24-s − 0.693·26-s + 1.45·27-s + 0.907·28-s − 1.01·29-s + 1.46·31-s − 1.77·32-s + 0.135·33-s − 2.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.804150043\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804150043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 11 | \( 1 - 0.264T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 - 8.13T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 3.95T + 61T^{2} \) |
| 67 | \( 1 - 9.20T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 + 7.93T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.445182316428219312371242006326, −7.927250568038951319947381302161, −7.56505706934959218595332645539, −6.71017307985038029685802318277, −5.84976200708260753823331619629, −4.37602062931772859839988380173, −3.37768034445879348746987076582, −2.60777444004819494132695128487, −1.85036518241716423618512677044, −0.984012084437720631301636677887,
0.984012084437720631301636677887, 1.85036518241716423618512677044, 2.60777444004819494132695128487, 3.37768034445879348746987076582, 4.37602062931772859839988380173, 5.84976200708260753823331619629, 6.71017307985038029685802318277, 7.56505706934959218595332645539, 7.927250568038951319947381302161, 8.445182316428219312371242006326