L(s) = 1 | − 3-s + 9-s + 2·13-s − 6·17-s + 8·19-s − 27-s + 6·29-s − 4·31-s + 10·37-s − 2·39-s + 6·41-s − 4·43-s + 6·51-s + 6·53-s − 8·57-s − 12·59-s + 10·61-s − 4·67-s − 12·71-s − 10·73-s − 8·79-s + 81-s − 12·83-s − 6·87-s + 6·89-s + 4·93-s − 10·97-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 1.83·19-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 + 0.840·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 1.28·61-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 − 0.643·87-s + 0.635·89-s + 0.414·93-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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
−14.63204893917950, −13.88802889567020, −13.65707992228726, −12.96209777779411, −12.70830759097566, −11.87293298038230, −11.40939177916668, −11.30722209879844, −10.45029747258597, −10.13485901036093, −9.366604314275756, −9.045681606976739, −8.399856582387103, −7.733298148206723, −7.218008504833096, −6.729835711503225, −6.014764337388920, −5.721561196077789, −4.939168582961961, −4.431789859813702, −3.899728349307960, −3.020099945320129, −2.541026512875468, −1.517765263434227, −0.9705024811466351, 0,
0.9705024811466351, 1.517765263434227, 2.541026512875468, 3.020099945320129, 3.899728349307960, 4.431789859813702, 4.939168582961961, 5.721561196077789, 6.014764337388920, 6.729835711503225, 7.218008504833096, 7.733298148206723, 8.399856582387103, 9.045681606976739, 9.366604314275756, 10.13485901036093, 10.45029747258597, 11.30722209879844, 11.40939177916668, 11.87293298038230, 12.70830759097566, 12.96209777779411, 13.65707992228726, 13.88802889567020, 14.63204893917950