| L(s) = 1 | − 3·3-s + 11·7-s + 9·9-s + 13-s + 37·19-s − 33·21-s − 27·27-s + 13·31-s − 26·37-s − 3·39-s − 61·43-s + 72·49-s − 111·57-s + 47·61-s + 99·63-s − 109·67-s + 46·73-s + 142·79-s + 81·81-s + 11·91-s − 39·93-s + 169·97-s + 194·103-s + 143·109-s + 78·111-s + 9·117-s + ⋯ |
| L(s) = 1 | − 3-s + 11/7·7-s + 9-s + 1/13·13-s + 1.94·19-s − 1.57·21-s − 27-s + 0.419·31-s − 0.702·37-s − 0.0769·39-s − 1.41·43-s + 1.46·49-s − 1.94·57-s + 0.770·61-s + 11/7·63-s − 1.62·67-s + 0.630·73-s + 1.79·79-s + 81-s + 0.120·91-s − 0.419·93-s + 1.74·97-s + 1.88·103-s + 1.31·109-s + 0.702·111-s + 1/13·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.816887516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816887516\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 13 T + p^{2} T^{2} \) |
| 37 | \( 1 + 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 61 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 + 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 46 T + p^{2} T^{2} \) |
| 79 | \( 1 - 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 169 T + p^{2} T^{2} \) |
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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
−9.742687072553244988898869514354, −8.678271658687009913752689204045, −7.75219083354338075609772339158, −7.17990359350676183934145909103, −6.06879174504315344422251618957, −5.13377780175133497107038966654, −4.77216709941841640505206344002, −3.50132755020527097741798501162, −1.85479285183852898729736294079, −0.910749993491308676555665672975,
0.910749993491308676555665672975, 1.85479285183852898729736294079, 3.50132755020527097741798501162, 4.77216709941841640505206344002, 5.13377780175133497107038966654, 6.06879174504315344422251618957, 7.17990359350676183934145909103, 7.75219083354338075609772339158, 8.678271658687009913752689204045, 9.742687072553244988898869514354
