L(s) = 1 | − i·2-s − 4-s − 5-s + i·8-s + i·10-s + 2.94i·11-s − 3.93i·13-s + 16-s + 0.399·17-s − 0.0352i·19-s + 20-s + 2.94·22-s − 3.73i·23-s + 25-s − 3.93·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 0.353i·8-s + 0.316i·10-s + 0.888i·11-s − 1.09i·13-s + 0.250·16-s + 0.0969·17-s − 0.00809i·19-s + 0.223·20-s + 0.628·22-s − 0.778i·23-s + 0.200·25-s − 0.771·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.254625288\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254625288\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 3.93iT - 13T^{2} \) |
| 17 | \( 1 - 0.399T + 17T^{2} \) |
| 19 | \( 1 + 0.0352iT - 19T^{2} \) |
| 23 | \( 1 + 3.73iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 0.828iT - 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 + 3.03T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 + 4.29iT - 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 1.93iT - 71T^{2} \) |
| 73 | \( 1 + 0.343iT - 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
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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
−8.549867004439504622232454439751, −7.61910428447702965898322412003, −7.13630919738869108267358632840, −6.10419977805095640044012360322, −5.15707034359622298852739336614, −4.62326474860446872595937317688, −3.64974512003229061848499742548, −2.97900538306499777189775334759, −1.99639120302288803500932174603, −0.863926167952210769702877552959,
0.46229113816633583786521485453, 1.83131717904016144191399823243, 3.11681623603594839636748147172, 3.93892506511177919234689972194, 4.59449169655293770030072298143, 5.55490891405378367758492392677, 6.19093948512183628525351248334, 6.88639362649545791904503987396, 7.69871816835052441100673251931, 8.151305739833295362056700458679