| L(s) = 1 | − 2-s + 4-s + 3.88·5-s − 3.72·7-s − 8-s − 3.88·10-s − 5.50·13-s + 3.72·14-s + 16-s + 4.37·17-s − 4.37·19-s + 3.88·20-s + 0.169·23-s + 10.1·25-s + 5.50·26-s − 3.72·28-s + 8.39·29-s − 0.337·31-s − 32-s − 4.37·34-s − 14.4·35-s + 1.89·37-s + 4.37·38-s − 3.88·40-s + 4.97·41-s + 3.84·43-s − 0.169·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.73·5-s − 1.40·7-s − 0.353·8-s − 1.23·10-s − 1.52·13-s + 0.995·14-s + 0.250·16-s + 1.06·17-s − 1.00·19-s + 0.869·20-s + 0.0352·23-s + 2.02·25-s + 1.08·26-s − 0.703·28-s + 1.55·29-s − 0.0605·31-s − 0.176·32-s − 0.749·34-s − 2.44·35-s + 0.312·37-s + 0.709·38-s − 0.615·40-s + 0.777·41-s + 0.585·43-s − 0.0249·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.460721208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.460721208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 - 0.169T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 0.337T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 + 2.69T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.016168702004983561008392745156, −7.24486143542403563246324845952, −6.40603514628662419236077976215, −6.22657876915250068344919838538, −5.37513167990121979150626840252, −4.55085818455696032503044751736, −3.14245108322288586055233993194, −2.64334773900199972851855501081, −1.87417789572046058469798122801, −0.67439926252964788415919261004,
0.67439926252964788415919261004, 1.87417789572046058469798122801, 2.64334773900199972851855501081, 3.14245108322288586055233993194, 4.55085818455696032503044751736, 5.37513167990121979150626840252, 6.22657876915250068344919838538, 6.40603514628662419236077976215, 7.24486143542403563246324845952, 8.016168702004983561008392745156
