L(s) = 1 | − 2.64·2-s + 0.547·3-s + 5.01·4-s − 1.45·6-s + 7-s − 7.98·8-s − 2.69·9-s − 6.33·11-s + 2.74·12-s + 4.94·13-s − 2.64·14-s + 11.1·16-s − 3.24·17-s + 7.14·18-s − 3.30·19-s + 0.547·21-s + 16.7·22-s − 23-s − 4.37·24-s − 13.1·26-s − 3.12·27-s + 5.01·28-s + 4.85·29-s − 0.831·31-s − 13.4·32-s − 3.46·33-s + 8.59·34-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 0.316·3-s + 2.50·4-s − 0.592·6-s + 0.377·7-s − 2.82·8-s − 0.899·9-s − 1.90·11-s + 0.792·12-s + 1.37·13-s − 0.707·14-s + 2.77·16-s − 0.786·17-s + 1.68·18-s − 0.758·19-s + 0.119·21-s + 3.57·22-s − 0.208·23-s − 0.892·24-s − 2.56·26-s − 0.601·27-s + 0.947·28-s + 0.901·29-s − 0.149·31-s − 2.37·32-s − 0.603·33-s + 1.47·34-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
\(L(1)\) |
\(\approx\) |
\(0.5227520958\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5227520958\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 0.547T + 3T^{2} \) |
| 11 | \( 1 + 6.33T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 0.831T + 31T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 8.84T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 + 1.15T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 8.99T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.451390098060742520025015065290, −8.086823405612434335403193971858, −7.38662898344270480338463010306, −6.39373366689595867245202768690, −5.87905684609567676302467089484, −4.80705097499991946583323737642, −3.34430467358621823077033650555, −2.54546842699312706650986463842, −1.87144215725312911099267246060, −0.51367215656297101092337929418,
0.51367215656297101092337929418, 1.87144215725312911099267246060, 2.54546842699312706650986463842, 3.34430467358621823077033650555, 4.80705097499991946583323737642, 5.87905684609567676302467089484, 6.39373366689595867245202768690, 7.38662898344270480338463010306, 8.086823405612434335403193971858, 8.451390098060742520025015065290