| L(s) = 1 | − 2.30·2-s + 1.95·3-s + 3.33·4-s + 2.21·5-s − 4.52·6-s + 4.98·7-s − 3.08·8-s + 0.833·9-s − 5.11·10-s − 1.08·11-s + 6.52·12-s − 5.85·13-s − 11.5·14-s + 4.33·15-s + 0.449·16-s − 3.36·17-s − 1.92·18-s + 4.22·19-s + 7.38·20-s + 9.75·21-s + 2.51·22-s + 23-s − 6.03·24-s − 0.0993·25-s + 13.5·26-s − 4.24·27-s + 16.6·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.13·3-s + 1.66·4-s + 0.990·5-s − 1.84·6-s + 1.88·7-s − 1.08·8-s + 0.277·9-s − 1.61·10-s − 0.328·11-s + 1.88·12-s − 1.62·13-s − 3.07·14-s + 1.11·15-s + 0.112·16-s − 0.815·17-s − 0.453·18-s + 0.968·19-s + 1.65·20-s + 2.12·21-s + 0.536·22-s + 0.208·23-s − 1.23·24-s − 0.0198·25-s + 2.65·26-s − 0.816·27-s + 3.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007044687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007044687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.95T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 - 4.98T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 + 5.85T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 - 4.59T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + 7.01T + 47T^{2} \) |
| 53 | \( 1 - 0.967T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 + 4.37T + 61T^{2} \) |
| 67 | \( 1 - 2.31T + 67T^{2} \) |
| 71 | \( 1 - 8.97T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 5.32T + 79T^{2} \) |
| 83 | \( 1 + 2.60T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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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
−8.012140242316328983994450233783, −7.52983750607799847953336759215, −6.97378851356620075057069874714, −5.84307388927366123467485352885, −5.00858077190490031856443146683, −4.42354014075571461209891376983, −2.87069205895357402774631831034, −2.22813326411573711520603534643, −1.91981949407008112559963063792, −0.856912614425409523136578274240,
0.856912614425409523136578274240, 1.91981949407008112559963063792, 2.22813326411573711520603534643, 2.87069205895357402774631831034, 4.42354014075571461209891376983, 5.00858077190490031856443146683, 5.84307388927366123467485352885, 6.97378851356620075057069874714, 7.52983750607799847953336759215, 8.012140242316328983994450233783
