| L(s) = 1 | + 0.414i·3-s + 2.82·9-s − 3.82·11-s − 3.58i·13-s + 6.41i·17-s − 3.65·19-s − 0.585i·23-s + 2.41i·27-s − 6.65·29-s + 4.58·31-s − 1.58i·33-s − 3.41i·37-s + 1.48·39-s + 0.585·41-s − 11.6i·43-s + ⋯ |
| L(s) = 1 | + 0.239i·3-s + 0.942·9-s − 1.15·11-s − 0.994i·13-s + 1.55i·17-s − 0.838·19-s − 0.122i·23-s + 0.464i·27-s − 1.23·29-s + 0.823·31-s − 0.276i·33-s − 0.561i·37-s + 0.237·39-s + 0.0914·41-s − 1.77i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7717060493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7717060493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 3.58iT - 13T^{2} \) |
| 17 | \( 1 - 6.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 0.585iT - 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 - 0.585T + 41T^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 + 8.89iT - 47T^{2} \) |
| 53 | \( 1 - 3.75iT - 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 5.17iT - 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−8.076683490449707361732007341644, −7.39596414822856554624628987604, −6.61187269646203785852972695368, −5.70576846458844858462004751733, −5.19510688623688580588751029599, −4.17434418845400546092575243295, −3.64173114974372782860567667977, −2.51347688857891057953767325495, −1.65624016750771714533787671191, −0.20771898525804678212672494509,
1.21231746158239985574459604730, 2.25687880044803200341387417381, 2.98975783680843786436648771087, 4.24360039677700922606545564939, 4.68084850300997118565488168304, 5.56219632750355922225812287448, 6.48827757450143749697379999194, 7.09524266260432613283493754652, 7.68809745054275504670915760011, 8.370699569487788367192065228200
