L(s) = 1 | − 2-s + (1 + i)3-s + 4-s + (−1 − i)6-s − 8-s + i·9-s + (1 + i)11-s + (1 + i)12-s + 16-s − 17-s − i·18-s + 2i·19-s + (−1 − i)22-s + (−1 − i)24-s − 32-s + 2i·33-s + ⋯ |
L(s) = 1 | − 2-s + (1 + i)3-s + 4-s + (−1 − i)6-s − 8-s + i·9-s + (1 + i)11-s + (1 + i)12-s + 16-s − 17-s − i·18-s + 2i·19-s + (−1 − i)22-s + (−1 − i)24-s − 32-s + 2i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.149947754\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.149947754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−9.032883690035726614863344845406, −8.473820128804237508209889326189, −7.80780056785084087366071227441, −6.92833690519049600528407255557, −6.26306008645796420053645587707, −5.12507247180082547333880838115, −3.99205179042792409060250874781, −3.60220289743565585917969749479, −2.38056493411393368941019534902, −1.63803324554598196518144561821,
0.844557040083936034175625390919, 1.86775558845767054019636798621, 2.73230140741530695782723455914, 3.38807882715577046615854375165, 4.70524820031684273215584093117, 6.06127419978028819299755566357, 6.71311567273992455936203853584, 7.12866398866948493251909583956, 8.010109674442084828369420628430, 8.652315077141616241069520373513