| L(s) = 1 | − i·2-s − 1.76·3-s − 4-s + 1.76i·6-s − 1.22·7-s + i·8-s + 0.105·9-s + 1.87·11-s + 1.76·12-s − 6.50i·13-s + 1.22i·14-s + 16-s − 0.765i·17-s − 0.105i·18-s + 3.34i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.01·3-s − 0.5·4-s + 0.719i·6-s − 0.461·7-s + 0.353i·8-s + 0.0350·9-s + 0.564·11-s + 0.508·12-s − 1.80i·13-s + 0.326i·14-s + 0.250·16-s − 0.185i·17-s − 0.0247i·18-s + 0.767i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02577407951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02577407951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + (5.82 - 1.76i)T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 6.50iT - 13T^{2} \) |
| 17 | \( 1 + 0.765iT - 17T^{2} \) |
| 19 | \( 1 - 3.34iT - 19T^{2} \) |
| 23 | \( 1 + 1.38iT - 23T^{2} \) |
| 29 | \( 1 + 1.72iT - 29T^{2} \) |
| 31 | \( 1 + 4.11iT - 31T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 4.91iT - 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 - 10.5iT - 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.963T + 71T^{2} \) |
| 73 | \( 1 + 9.03T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 0.00656T + 83T^{2} \) |
| 89 | \( 1 + 4.70iT - 89T^{2} \) |
| 97 | \( 1 + 0.403iT - 97T^{2} \) |
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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
−8.769798873495916895786843759129, −8.030848829871158837590545270322, −7.01820430278563111735032639301, −5.93079571557213679497965230381, −5.61476706870883160934233282393, −4.56984106332019415163690410559, −3.52591220426456534798959703970, −2.67850628498119027050192120697, −1.12481632234319780636714048071, −0.01269401068764210148265249810,
1.59923341294373596170714733381, 3.24827747808467480756753250762, 4.36742179527510075209926552239, 5.01692112310341755820489586048, 5.97945510662353000390782571679, 6.68759949070703740520866194397, 6.95928039118913776369560247614, 8.228986931646784266436162913699, 9.118035634536777548070994623704, 9.510533477588867974766508673855
