L(s) = 1 | − 2-s − 1.80·3-s + 4-s + 5-s + 1.80·6-s − 3.60·7-s − 8-s + 0.246·9-s − 10-s + 4.44·11-s − 1.80·12-s + 3.60·14-s − 1.80·15-s + 16-s − 3.15·17-s − 0.246·18-s − 0.356·19-s + 20-s + 6.49·21-s − 4.44·22-s − 6.49·23-s + 1.80·24-s + 25-s + 4.96·27-s − 3.60·28-s − 2.89·29-s + 1.80·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.447·5-s + 0.735·6-s − 1.36·7-s − 0.353·8-s + 0.0823·9-s − 0.316·10-s + 1.34·11-s − 0.520·12-s + 0.963·14-s − 0.465·15-s + 0.250·16-s − 0.766·17-s − 0.0582·18-s − 0.0818·19-s + 0.223·20-s + 1.41·21-s − 0.947·22-s − 1.35·23-s + 0.367·24-s + 0.200·25-s + 0.954·27-s − 0.681·28-s − 0.536·29-s + 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5839953186\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5839953186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + 0.356T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 2.03T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 6.96T + 73T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 - 3.95T + 97T^{2} \) |
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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
−9.289690976103619557007255805828, −8.942764106038748322941569958008, −7.68131644915018153669583701310, −6.53060448619932686868603863775, −6.40518576411704686942637735988, −5.67350476282372621730028534309, −4.36273811174029108111465395417, −3.30842072805832387480638305107, −2.03832017204313545327254565069, −0.60036263492667065419793626306,
0.60036263492667065419793626306, 2.03832017204313545327254565069, 3.30842072805832387480638305107, 4.36273811174029108111465395417, 5.67350476282372621730028534309, 6.40518576411704686942637735988, 6.53060448619932686868603863775, 7.68131644915018153669583701310, 8.942764106038748322941569958008, 9.289690976103619557007255805828