| L(s) = 1 | − 2.61i·3-s − 1.08·5-s + (1.08 − 2.41i)7-s − 3.82·9-s − 2·11-s − 4.14·13-s + 2.82i·15-s − 7.39i·17-s + 4.77i·19-s + (−6.30 − 2.82i)21-s + 3.65i·23-s − 3.82·25-s + 2.16i·27-s + 7.65i·29-s + 7.39·31-s + ⋯ |
| L(s) = 1 | − 1.50i·3-s − 0.484·5-s + (0.409 − 0.912i)7-s − 1.27·9-s − 0.603·11-s − 1.14·13-s + 0.730i·15-s − 1.79i·17-s + 1.09i·19-s + (−1.37 − 0.617i)21-s + 0.762i·23-s − 0.765·25-s + 0.416i·27-s + 1.42i·29-s + 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4218968896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4218968896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.08 + 2.41i)T \) |
| good | 3 | \( 1 + 2.61iT - 3T^{2} \) |
| 5 | \( 1 + 1.08T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 + 7.39iT - 17T^{2} \) |
| 19 | \( 1 - 4.77iT - 19T^{2} \) |
| 23 | \( 1 - 3.65iT - 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 8.28iT - 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 5.67iT - 59T^{2} \) |
| 61 | \( 1 + 1.08T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 - 3.17iT - 71T^{2} \) |
| 73 | \( 1 - 0.896iT - 73T^{2} \) |
| 79 | \( 1 - 7.17iT - 79T^{2} \) |
| 83 | \( 1 - 1.71iT - 83T^{2} \) |
| 89 | \( 1 - 5.22iT - 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−8.233900993550541123501746583004, −7.86296016377743744095575383111, −7.13506708544196717258364553315, −6.81197063376808938637363049028, −5.45546742813685076620430452374, −4.76677731973686489301678455553, −3.47472829690090829505732489918, −2.42581230978839123383350827656, −1.33505109653191621754103155181, −0.15652116759783073942992291490,
2.23481821354497266677072801181, 3.11475346541749140859630314224, 4.32588556836758119789008160004, 4.67524530074877657135559951026, 5.58775337872355216082777934082, 6.43407216965773642602693679237, 7.82544518115152590578955268561, 8.313434815750120267511830943905, 9.109814983452115798989260850998, 9.938424470601601998060878837526
