| L(s) = 1 | − 2.61·3-s + 2.85·7-s + 3.85·9-s − 11-s + 6.23·13-s − 0.618·17-s + 6.70·19-s − 7.47·21-s + 4.09·23-s − 2.23·27-s − 1.38·29-s + 3·31-s + 2.61·33-s + 10.2·37-s − 16.3·39-s − 3·41-s + 6·43-s − 11.9·47-s + 1.14·49-s + 1.61·51-s + 9.32·53-s − 17.5·57-s − 0.527·59-s + 0.0901·61-s + 11.0·63-s − 8·67-s − 10.7·69-s + ⋯ |
| L(s) = 1 | − 1.51·3-s + 1.07·7-s + 1.28·9-s − 0.301·11-s + 1.72·13-s − 0.149·17-s + 1.53·19-s − 1.63·21-s + 0.852·23-s − 0.430·27-s − 0.256·29-s + 0.538·31-s + 0.455·33-s + 1.68·37-s − 2.61·39-s − 0.468·41-s + 0.914·43-s − 1.74·47-s + 0.163·49-s + 0.226·51-s + 1.28·53-s − 2.32·57-s − 0.0687·59-s + 0.0115·61-s + 1.38·63-s − 0.977·67-s − 1.28·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542263610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542263610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 0.527T + 59T^{2} \) |
| 61 | \( 1 - 0.0901T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.85T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−8.234606243764627901304360733427, −7.60764956526130263936496252488, −6.73593281513579639663115836049, −6.03837181311176266923668753874, −5.42550118355934300981434878868, −4.85519427458334599103766229175, −4.06067590764984919338574741528, −2.96302526302193951353364385140, −1.46934169082693586067313946329, −0.862705828975303626260603419420,
0.862705828975303626260603419420, 1.46934169082693586067313946329, 2.96302526302193951353364385140, 4.06067590764984919338574741528, 4.85519427458334599103766229175, 5.42550118355934300981434878868, 6.03837181311176266923668753874, 6.73593281513579639663115836049, 7.60764956526130263936496252488, 8.234606243764627901304360733427
