| L(s) = 1 | + 3·3-s − 5-s + 6·9-s + 2·11-s − 6·13-s − 3·15-s + 2·17-s + 9·23-s + 25-s + 9·27-s + 3·29-s − 2·31-s + 6·33-s + 8·37-s − 18·39-s + 5·41-s − 43-s − 6·45-s − 8·47-s + 6·51-s + 4·53-s − 2·55-s + 8·59-s + 7·61-s + 6·65-s + 3·67-s + 27·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s + 0.603·11-s − 1.66·13-s − 0.774·15-s + 0.485·17-s + 1.87·23-s + 1/5·25-s + 1.73·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s + 1.31·37-s − 2.88·39-s + 0.780·41-s − 0.152·43-s − 0.894·45-s − 1.16·47-s + 0.840·51-s + 0.549·53-s − 0.269·55-s + 1.04·59-s + 0.896·61-s + 0.744·65-s + 0.366·67-s + 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.552832436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.552832436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.445531946071457035342758439215, −7.78747984091549074394747892658, −7.23578016302986501702556097702, −6.65281836855627160860891446656, −5.21800614555602391782680540299, −4.50044027970320622351460760500, −3.68444220862026189222115915519, −2.90295502645666231695581641474, −2.30755956725717539613876163361, −1.04538800149554692365529434659,
1.04538800149554692365529434659, 2.30755956725717539613876163361, 2.90295502645666231695581641474, 3.68444220862026189222115915519, 4.50044027970320622351460760500, 5.21800614555602391782680540299, 6.65281836855627160860891446656, 7.23578016302986501702556097702, 7.78747984091549074394747892658, 8.445531946071457035342758439215
