| L(s) = 1 | + 2.72·3-s − 1.08i·5-s + 4.41·9-s + 2.08i·11-s + 2.61i·13-s − 2.94i·15-s − 4.46i·17-s − 1.12·19-s − 7.11i·23-s + 3.82·25-s + 3.85·27-s + 1.17·29-s + 7.70·31-s + 5.67i·33-s + 4·37-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.484i·5-s + 1.47·9-s + 0.628i·11-s + 0.724i·13-s − 0.760i·15-s − 1.08i·17-s − 0.258·19-s − 1.48i·23-s + 0.765·25-s + 0.741·27-s + 0.217·29-s + 1.38·31-s + 0.987i·33-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.456287923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.456287923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.72T + 3T^{2} \) |
| 5 | \( 1 + 1.08iT - 5T^{2} \) |
| 11 | \( 1 - 2.08iT - 11T^{2} \) |
| 13 | \( 1 - 2.61iT - 13T^{2} \) |
| 17 | \( 1 + 4.46iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 5.54iT - 41T^{2} \) |
| 43 | \( 1 + 7.97iT - 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 13.0iT - 61T^{2} \) |
| 67 | \( 1 - 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 7.97iT - 73T^{2} \) |
| 79 | \( 1 - 4.16iT - 79T^{2} \) |
| 83 | \( 1 + 4.31T + 83T^{2} \) |
| 89 | \( 1 - 4.01iT - 89T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−8.750129413957297150425724487861, −8.121156652201416760280432746679, −7.13271049066152064916340656197, −6.79512815286898542705800441943, −5.43151850849573211804286599860, −4.42590851624083968865010556700, −4.04473671878837788123315746570, −2.66453851019187254175272130088, −2.38365577607367737412148308850, −1.00547970102265980263832673278,
1.23252509175208163959609707858, 2.39767946479423931207129493325, 3.14137315863814565354305499500, 3.66560412572818464436612187728, 4.67342023888255030453698047773, 5.83903666533105245605935379369, 6.56463963629698042654849188924, 7.54451834552028106499872907894, 8.107368295717380144897076398823, 8.543253030395735737319884483617
