| L(s) = 1 | + 5.38·3-s − i·7-s + 19.9·9-s + 14·11-s + 16.1·13-s + 23i·17-s + (−10 − 16.1i)19-s − 5.38i·21-s + i·23-s + 59.2·27-s + 48.4i·29-s − 32.3i·31-s + 75.3·33-s − 32.3·37-s + 86.9·39-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.142i·7-s + 2.22·9-s + 1.27·11-s + 1.24·13-s + 1.35i·17-s + (−0.526 − 0.850i)19-s − 0.256i·21-s + 0.0434i·23-s + 2.19·27-s + 1.67i·29-s − 1.04i·31-s + 2.28·33-s − 0.873·37-s + 2.23·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.838890508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.838890508\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (10 + 16.1i)T \) |
| good | 3 | \( 1 - 5.38T + 9T^{2} \) |
| 7 | \( 1 + iT - 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 - 16.1T + 169T^{2} \) |
| 17 | \( 1 - 23iT - 289T^{2} \) |
| 23 | \( 1 - iT - 529T^{2} \) |
| 29 | \( 1 - 48.4iT - 841T^{2} \) |
| 31 | \( 1 + 32.3iT - 961T^{2} \) |
| 37 | \( 1 + 32.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 26iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 80.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 96.9T + 9.40e3T^{2} \) |
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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
−8.818387476454031230315393968226, −8.574471111547719819239975432246, −7.64548214881492823881259366063, −6.80507036719030699884244511215, −6.11050817206310978634847086898, −4.57992680007553200580072990031, −3.75162765688387679066940260592, −3.33874858115252401988410218141, −2.01987214720837216679695337451, −1.29443436105305469586945536014,
1.18865511936312330932264021290, 2.10587540819028764995550021786, 3.17348118886089477177487667295, 3.80694714752311074483267923994, 4.57855202345696301591488072292, 6.02027850387531600604936152991, 6.81980249228743873162588314197, 7.64429279372577691618460730225, 8.432379906625334531529872860888, 8.907461305484685276163437957525
