L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 + i)19-s + 1.41·23-s + 2i·31-s + (−0.707 − 0.707i)32-s + (1.00 + 1.00i)34-s − 1.41·38-s + (1.00 + 1.00i)46-s + 1.41i·47-s − 49-s + (−1 − i)61-s + (−1.41 + 1.41i)62-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 + i)19-s + 1.41·23-s + 2i·31-s + (−0.707 − 0.707i)32-s + (1.00 + 1.00i)34-s − 1.41·38-s + (1.00 + 1.00i)46-s + 1.41i·47-s − 49-s + (−1 − i)61-s + (−1.41 + 1.41i)62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.788782711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788782711\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 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
−8.739548244820448964306553226188, −8.064195037630132165942152590804, −7.44391627976619309053602629073, −6.63365072542530337417233310140, −6.00477529917657942880727165907, −5.17550548795675326349954246620, −4.57505655808519907944404620231, −3.49023799246816021736920406591, −2.98954593424674399346939061585, −1.56873612633801131118794968377,
0.864490264831248719013907376912, 2.11342649451809806666003523781, 2.96059460854360747427595305849, 3.78500489946826328544568450396, 4.64682768377721411346024987088, 5.33182694746686031333768168516, 6.10727422809373159310536179858, 6.86213226580840202940040756303, 7.69219125961290442065247765548, 8.677544458126776470747749793296