L(s) = 1 | + 3.36i·3-s − 1.90i·7-s − 8.28·9-s − 5.48·11-s + 1.04i·13-s − 6.74i·17-s + 1.55·19-s + 6.40·21-s + i·23-s − 17.7i·27-s + 3.38·29-s + 10.9·31-s − 18.4i·33-s + 5.26i·37-s − 3.52·39-s + ⋯ |
L(s) = 1 | + 1.93i·3-s − 0.720i·7-s − 2.76·9-s − 1.65·11-s + 0.291i·13-s − 1.63i·17-s + 0.355·19-s + 1.39·21-s + 0.208i·23-s − 3.42i·27-s + 0.628·29-s + 1.96·31-s − 3.20i·33-s + 0.865i·37-s − 0.564·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.326022683\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.326022683\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 3.36iT - 3T^{2} \) |
| 7 | \( 1 + 1.90iT - 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 - 1.04iT - 13T^{2} \) |
| 17 | \( 1 + 6.74iT - 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 5.26iT - 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 0.403iT - 47T^{2} \) |
| 53 | \( 1 + 5.88iT - 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 - 0.478T + 71T^{2} \) |
| 73 | \( 1 + 2.40iT - 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−8.484476663745345247720153675219, −7.950313550903037592950196986886, −7.06367327801797111591808747168, −5.98757936276875519810773182868, −5.17041939636542876034685845224, −4.72986573413002767087353325779, −4.13929714621833268716539792401, −2.93120177236798578475703339484, −2.77983141022125751479333583831, −0.55996922729282547546059407866,
0.68577344853494979427635592576, 1.80869219076614632485252519508, 2.56699157076202599025529859175, 3.10562576866720524314307466914, 4.66667543551575697547531484529, 5.71780258066526122962758748177, 5.95570794922934059607598199005, 6.73657885490260994775681118669, 7.68354961758836245006548836140, 8.023483773802500380438672701928