L(s) = 1 | − i·2-s − 4-s + (0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 − 0.707i)10-s + 1.41·11-s + 16-s + i·18-s + 1.41·19-s + (−0.707 − 0.707i)20-s − 1.41i·22-s − i·23-s + 1.00i·25-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (0.707 + 0.707i)5-s + i·8-s − 9-s + (0.707 − 0.707i)10-s + 1.41·11-s + 16-s + i·18-s + 1.41·19-s + (−0.707 − 0.707i)20-s − 1.41i·22-s − i·23-s + 1.00i·25-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.030496807\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.030496807\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.29412322961486256777453383132, −9.349149821510299132500395570967, −8.960494878923566746990570832643, −7.82017834391380506956818001553, −6.58187428643003173866147719091, −5.80844023428620525715278782201, −4.77131863069089263679239149411, −3.48506793406658032330584439213, −2.78080629863118656953457598317, −1.48864566008807305982573958157,
1.33406800955825645230388807727, 3.26105392118027306072406529125, 4.40232285131080856208065065126, 5.46705593914220961583873457589, 5.93174416523795499663876520905, 6.90887890549995765809552213121, 7.890754425727417732224881711797, 8.797480320733567657256113004715, 9.335451960216499489175210828685, 9.900854354672075870727268596045