| L(s) = 1 | − 2·7-s + 11-s + 4·13-s + 6·17-s + 8·19-s − 3·23-s + 5·31-s + 37-s + 10·43-s − 3·49-s − 6·53-s − 3·59-s − 4·61-s + 67-s − 15·71-s + 4·73-s − 2·77-s + 2·79-s + 6·83-s + 9·89-s − 8·91-s + 7·97-s − 18·101-s − 8·103-s + 6·107-s + 2·109-s − 15·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 0.898·31-s + 0.164·37-s + 1.52·43-s − 3/7·49-s − 0.824·53-s − 0.390·59-s − 0.512·61-s + 0.122·67-s − 1.78·71-s + 0.468·73-s − 0.227·77-s + 0.225·79-s + 0.658·83-s + 0.953·89-s − 0.838·91-s + 0.710·97-s − 1.79·101-s − 0.788·103-s + 0.580·107-s + 0.191·109-s − 1.41·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.365672233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365672233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.72377070454418064753635594980, −7.00384763041942808180128597508, −6.10999373971084019596899432654, −5.84911857242065243215267609867, −4.97962786716775383120380458870, −4.02434365601982571256571585661, −3.34352896754938347295668800791, −2.87236229366036766390783361878, −1.50686586618168342155746753413, −0.796223665826135475945172747678,
0.796223665826135475945172747678, 1.50686586618168342155746753413, 2.87236229366036766390783361878, 3.34352896754938347295668800791, 4.02434365601982571256571585661, 4.97962786716775383120380458870, 5.84911857242065243215267609867, 6.10999373971084019596899432654, 7.00384763041942808180128597508, 7.72377070454418064753635594980
