L(s) = 1 | + 2·3-s + 2·4-s + 4i·5-s − 2i·7-s + 9-s + i·11-s + 4·12-s + (−2 − 3i)13-s + 8i·15-s + 4·16-s − 2·17-s − 2i·19-s + 8i·20-s − 4i·21-s − 4·23-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 4-s + 1.78i·5-s − 0.755i·7-s + 0.333·9-s + 0.301i·11-s + 1.15·12-s + (−0.554 − 0.832i)13-s + 2.06i·15-s + 16-s − 0.485·17-s − 0.458i·19-s + 1.78i·20-s − 0.872i·21-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17962 + 0.659937i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17962 + 0.659937i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (2 + 3i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 4iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 15iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−11.11640169391882038432642624454, −10.40650301407362823743526944467, −9.865101921932921833798347143972, −8.330891792706934249857488501637, −7.39411504380701027906515046948, −7.03246694791877821157741815824, −5.93747167203652307881585637367, −3.91813201613630258886152375650, −2.87913545534096285272776893375, −2.33898214138781229335548978487,
1.67317628191153079059259795335, 2.67639293080402667122026786589, 4.15389208039854490017879517281, 5.34892375760208459795988448951, 6.42619710812602113355812489125, 7.87332931386543365109413609431, 8.424697193241376930589622367273, 9.148093643779623950381364834693, 9.950788799957586492306938655011, 11.48824188220044485522628631094