| L(s) = 1 | + (1 − i)2-s − 2i·4-s + 2i·5-s − 7-s + (−2 − 2i)8-s + (2 + 2i)10-s − 5i·13-s + (−1 + i)14-s − 4·16-s − 17-s − 4i·19-s + 4·20-s + 5·23-s + 25-s + (−5 − 5i)26-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + 0.894i·5-s − 0.377·7-s + (−0.707 − 0.707i)8-s + (0.632 + 0.632i)10-s − 1.38i·13-s + (−0.267 + 0.267i)14-s − 16-s − 0.242·17-s − 0.917i·19-s + 0.894·20-s + 1.04·23-s + 0.200·25-s + (−0.980 − 0.980i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.831719668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.831719668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.486860070190795751361069327113, −8.499748507469178522158671167291, −7.28543708643013612681969357116, −6.66589630943182756519649759599, −5.73641845894931392643644888595, −5.00284110450132716852263924173, −3.80587755456490075634161400073, −3.03793507479882231506280641068, −2.29847246792256366322253104066, −0.56071149998203201507387575933,
1.61535311617787532098263571570, 3.08612137389323668044996812436, 4.05826532266831872858464431295, 4.84020235400480978704004580600, 5.56201561497141430428530814003, 6.58304023680905080943255699155, 7.13500070362849251630449987816, 8.135183073746142513711169630680, 9.063914088281094391248173056872, 9.225278875551982205436574253455
