L(s) = 1 | + (−0.176 + 0.306i)3-s + 4.30·7-s + (1.43 + 2.48i)9-s + 6.01·11-s + (−2.97 − 5.15i)13-s + (1.93 − 3.35i)17-s + (4.19 + 1.17i)19-s + (−0.760 + 1.31i)21-s + (0.391 + 0.678i)23-s − 2.07·27-s + (−3.98 − 6.89i)29-s − 4.49·31-s + (−1.06 + 1.84i)33-s + 0.988·37-s + 2.10·39-s + ⋯ |
L(s) = 1 | + (−0.102 + 0.176i)3-s + 1.62·7-s + (0.479 + 0.829i)9-s + 1.81·11-s + (−0.825 − 1.42i)13-s + (0.469 − 0.813i)17-s + (0.963 + 0.268i)19-s + (−0.166 + 0.287i)21-s + (0.0816 + 0.141i)23-s − 0.399·27-s + (−0.739 − 1.28i)29-s − 0.806·31-s + (−0.184 + 0.320i)33-s + 0.162·37-s + 0.336·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.374451558\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.374451558\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.19 - 1.17i)T \) |
good | 3 | \( 1 + (0.176 - 0.306i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 - 6.01T + 11T^{2} \) |
| 13 | \( 1 + (2.97 + 5.15i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.391 - 0.678i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.98 + 6.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 0.988T + 37T^{2} \) |
| 41 | \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.785 - 1.35i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.630 + 1.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.07 + 7.05i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.62 - 4.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.80 + 4.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.52 - 6.09i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.90 + 5.03i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.62 - 8.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.99 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.58T + 83T^{2} \) |
| 89 | \( 1 + (-1.69 - 2.93i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.69 - 6.40i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.368784596541782353327339116880, −8.190249910359632009123816127010, −7.72501547675270635140438813438, −7.09750219006934941974534767655, −5.76196421480094750710800257255, −5.09772652737599972975825269469, −4.45467586822956731016574868253, −3.39446107394701473249464181313, −2.04135823160590681669107460058, −1.10266223427222463590687552903,
1.35277737591397920257655032417, 1.77110639019049965939970615945, 3.58226047882359449999838266827, 4.26650303086279748338769461326, 5.06942801496316408694755338277, 6.11485531674369764711261857491, 7.05195466584209128476320834926, 7.39294122448085751332443778932, 8.615665986997794545772718701224, 9.170814045389700872549909098751