L(s) = 1 | + 3-s + 7-s + 9-s − 2·13-s + 6·17-s − 8·19-s + 21-s + 27-s + 6·29-s + 4·31-s + 10·37-s − 2·39-s − 6·41-s − 4·43-s + 49-s + 6·51-s + 6·53-s − 8·57-s + 12·59-s − 10·61-s + 63-s − 4·67-s − 12·71-s + 10·73-s − 8·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 0.218·21-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.125·63-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.712530060\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.712530060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.922925278628559095007077965993, −7.21373516827613780263565835707, −6.44756599598651588851472927849, −5.78808945627965836574415359950, −4.80110370873327529342210539656, −4.35655340812170958925783475261, −3.40957197982514765056849415782, −2.64321891154009254076554460028, −1.87662685639268254821958237764, −0.792412202651869171388469681652,
0.792412202651869171388469681652, 1.87662685639268254821958237764, 2.64321891154009254076554460028, 3.40957197982514765056849415782, 4.35655340812170958925783475261, 4.80110370873327529342210539656, 5.78808945627965836574415359950, 6.44756599598651588851472927849, 7.21373516827613780263565835707, 7.922925278628559095007077965993