| L(s) = 1 | + i·2-s − 2.73i·3-s − 4-s + 2.73·6-s + i·7-s − i·8-s − 4.46·9-s + 11-s + 2.73i·12-s + 1.46i·13-s − 14-s + 16-s − 3.46i·17-s − 4.46i·18-s − 6.73·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.57i·3-s − 0.5·4-s + 1.11·6-s + 0.377i·7-s − 0.353i·8-s − 1.48·9-s + 0.301·11-s + 0.788i·12-s + 0.406i·13-s − 0.267·14-s + 0.250·16-s − 0.840i·17-s − 1.05i·18-s − 1.54·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127906208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127906208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73iT - 3T^{2} \) |
| 13 | \( 1 - 1.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 0.732iT - 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 7.26iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 14.3iT - 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−8.319995350117891061188429783215, −7.81224447918294722088954409150, −7.04810784971909090283014688097, −6.54215250150610590122403130556, −5.96905765013831786636805037301, −5.11170848095506488781952743094, −4.14599854660980328190164171188, −2.93300624718859933761900532654, −1.99836628076775742004816954794, −1.02984674606785596243924365371,
0.37272446864694396930898570592, 1.98799247020096320542397459294, 3.03712525212981341941203744998, 3.79786543890403608764078901730, 4.46878826686280481761640161156, 4.87925904715799489005187266624, 6.02394424276789619713875093028, 6.65976477124253615768297357434, 8.075826123785307371409970730214, 8.579797319910735342384716898411
