| L(s) = 1 | + 1.05·3-s − 1.97·5-s + 4.35·7-s − 1.89·9-s − 3.68·11-s − 3.40·13-s − 2.08·15-s + 0.913·17-s − 2.61·19-s + 4.58·21-s + 3.15·23-s − 1.07·25-s − 5.14·27-s + 9.69·29-s + 7.70·31-s − 3.87·33-s − 8.62·35-s + 0.206·37-s − 3.58·39-s + 8.61·41-s − 5.42·43-s + 3.74·45-s + 8.71·47-s + 11.9·49-s + 0.961·51-s − 2.86·53-s + 7.29·55-s + ⋯ |
| L(s) = 1 | + 0.607·3-s − 0.885·5-s + 1.64·7-s − 0.630·9-s − 1.11·11-s − 0.944·13-s − 0.538·15-s + 0.221·17-s − 0.599·19-s + 1.00·21-s + 0.658·23-s − 0.215·25-s − 0.991·27-s + 1.80·29-s + 1.38·31-s − 0.674·33-s − 1.45·35-s + 0.0340·37-s − 0.573·39-s + 1.34·41-s − 0.827·43-s + 0.558·45-s + 1.27·47-s + 1.70·49-s + 0.134·51-s − 0.393·53-s + 0.983·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.942406772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942406772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 - 0.913T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 - 9.69T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 0.206T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 5.42T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 + 1.93T + 71T^{2} \) |
| 73 | \( 1 - 7.38T + 73T^{2} \) |
| 79 | \( 1 - 0.551T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 0.889T + 97T^{2} \) |
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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
−8.457499663508001020853350546975, −7.71082971371521931688416668854, −7.48594821056148428163947557330, −6.23858635242848073882367516461, −5.11680844348316527206421685044, −4.79326797897042681311277617785, −3.89776116962614639791273756749, −2.72808012094288222109113286703, −2.28315316618354154060221031657, −0.76088005307399552919713448735,
0.76088005307399552919713448735, 2.28315316618354154060221031657, 2.72808012094288222109113286703, 3.89776116962614639791273756749, 4.79326797897042681311277617785, 5.11680844348316527206421685044, 6.23858635242848073882367516461, 7.48594821056148428163947557330, 7.71082971371521931688416668854, 8.457499663508001020853350546975
