L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + 2·11-s + 7i·13-s + 14-s + 16-s − 7i·17-s − 8·19-s + 2i·22-s + 5i·23-s − 7·26-s + i·28-s + 9·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s + 0.603·11-s + 1.94i·13-s + 0.267·14-s + 0.250·16-s − 1.69i·17-s − 1.83·19-s + 0.426i·22-s + 1.04i·23-s − 1.37·26-s + 0.188i·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.076347863\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076347863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 7iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 5iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 - 3iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.853275170807142537891153385401, −8.359063473281505591935829106303, −7.25166980335507169927531130144, −6.73277316906065924708901819067, −6.32957002532125918216990431556, −5.04970164333507718777609582642, −4.47297915487857364706196279247, −3.76511834172431116782694093027, −2.46935528906122070996103266726, −1.26326178822355029715871554354,
0.34920238595241510993526294456, 1.67568919804336700782523855238, 2.62588015808960914988669655533, 3.51648091045218432085558216112, 4.32464660042979915556858373212, 5.19463959774859308182043047669, 6.14202258076991458656565764398, 6.61436164253623202977620184795, 8.101684733450376567932985661289, 8.340033520648519684455229869048