L(s) = 1 | + (−0.951 + 0.690i)2-s + (−0.309 − 0.951i)3-s + (0.118 − 0.363i)4-s + (1.53 + 1.11i)5-s + (0.951 + 0.690i)6-s + (−0.224 − 0.690i)8-s + (−0.809 + 0.587i)9-s − 2.23·10-s + (−0.587 + 0.809i)11-s − 0.381·12-s + (0.809 − 0.587i)13-s + (0.587 − 1.80i)15-s + (0.999 + 0.726i)16-s + (0.363 − 1.11i)18-s + (0.587 − 0.427i)20-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.690i)2-s + (−0.309 − 0.951i)3-s + (0.118 − 0.363i)4-s + (1.53 + 1.11i)5-s + (0.951 + 0.690i)6-s + (−0.224 − 0.690i)8-s + (−0.809 + 0.587i)9-s − 2.23·10-s + (−0.587 + 0.809i)11-s − 0.381·12-s + (0.809 − 0.587i)13-s + (0.587 − 1.80i)15-s + (0.999 + 0.726i)16-s + (0.363 − 1.11i)18-s + (0.587 − 0.427i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5794626791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5794626791\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21767897935330953325198753794, −10.36383123347516927102883972620, −9.748273535454978068409905105211, −8.652615700436890498185208447759, −7.66967343643274434130268235562, −6.89856689822672206107379900204, −6.24896461850620929455597854968, −5.42748429448858997062667933421, −3.05892434731011445562152044693, −1.77684000250295276032352506307,
1.30437242615509866641677873868, 2.77380936128371061368570598996, 4.54004502837107880212446274385, 5.59053938126484071494115898510, 6.08292241057315267795428473586, 8.285656851553297498226264612011, 8.962458077530065192437702136688, 9.441544221589262475221545055156, 10.27313225566652335663711525976, 10.88108725916123347196898584287